View source for Linear operator

''linear transformation''

A mapping between two vector spaces (cf. [[Vector space|Vector space]]) that is compatible with their linear structures. More precisely, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593403.png" /> are vector spaces over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593404.png" />, is called a linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593405.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593406.png" /> if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593407.png" /></td> </tr></table>

for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593408.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l0593409.png" />. The simplest examples are the zero linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934010.png" />, which takes all vectors into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934011.png" />, and (in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934012.png" />) the identity linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934013.png" />, which leaves all vectors unchanged.

The concept of a linear operator, which together with the concept of a vector space is fundamental in linear algebra, plays a role in very diverse branches of mathematics and physics, above all in analysis and its applications.

The modern definition of a linear operator was first given by G. Peano [[#References|[1]]] (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934014.png" />). However, it was rooted in the previous developments of mathematics, which had accumulated (beginning with the linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934015.png" />) a vast number of examples. In algebra an incomplete list of them includes linear substitutions in systems of linear equations, and multiplication of quaternions and elements of a Grassmann algebra; in analytic geometry it includes coordinate transformations; in analysis it includes differential and integral transforms and the Fourier integral.

Up to the beginning of the 20th century the only linear operators that had been systematically studied were those between finite-dimensional spaces over the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934017.png" />. The first  "infinite-dimensional"  observations, concerned also with general fields, were made by O. Toeplitz [[#References|[3]]]. As a rule, linear operators between infinite-dimensional spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934019.png" /> are studied under the assumption that they are continuous with respect to certain topologies. Continuous linear operators that act in various classes of topological vector spaces, in the first place Banach and Hilbert spaces, are the main object of study of linear functional analysis (cf. also [[Banach space|Banach space]]; [[Hilbert space|Hilbert space]]; [[Topological vector space|Topological vector space]]).

In the theory of linear operators the two special cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934021.png" /> are the most important. In the first case a linear operator is called a functional (see [[Linear functional|Linear functional]]), and in the second case a linear operator acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934022.png" />, or an endomorphism.

The linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934023.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934024.png" /> form a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934025.png" /> (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934026.png" /> one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934027.png" />) over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934028.png" /> with respect to addition and multiplication by a scalar, given by the formulas

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934029.png" /></td> </tr></table>

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934031.png" />; the zero is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934032.png" />. Multiplication (composition) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934033.png" /> of linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934035.png" /> is defined only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934036.png" /> as the successive application of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934038.png" />. With respect to these three operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934039.png" /> is an example of an associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934040.png" /> with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934041.png" /> (cf. [[Associative rings and algebras|Associative rings and algebras]]). This is  "more than an example" : Every associative algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934042.png" /> can be imbedded in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934043.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934044.png" />.

Vector spaces over a fixed field (objects) and linear operators (morphisms) form, together with the composition law, the [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934045.png" />. The following concepts are special cases (in connection with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934046.png" />) of general categorical concepts. For a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934047.png" /> its kernel is the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934048.png" />, its image is the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934049.png" />, and its cokernel is the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934050.png" />. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934051.png" /> is called a monomorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934052.png" /> and an epimorphism if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934053.png" />. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934054.png" /> is called the left (respectively, right) inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934055.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934056.png" /> is the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934057.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934058.png" /> is the identity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934059.png" />). A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934060.png" /> that is simultaneously the left and right inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934061.png" /> is called the inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934062.png" />. A linear operator (respectively, endomorphism) that has an inverse is called an isomorphism (respectively, automorphism).

The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934063.png" /> is an [[Abelian category|Abelian category]] with respect to addition of linear operators; in particular, a linear operator that is a monomorphism and an epimorphism is an isomorphism. Moreover, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934064.png" /> every monomorphism has a left inverse and every epimorphism has a right inverse. By analogy with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934065.png" /> one introduces the categories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934066.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934067.png" />; the objects of the first are Banach spaces, and the objects of the second are Hilbert spaces; the morphisms in both are continuous linear operators. Both categories are additive (cf. [[Additive category|Additive category]]), but not Abelian. Isomorphisms in them are called topological isomorphisms; these are linear operators that have a continuous inverse.

One of the most important typical problems of the  "intrinsic"  theory of linear operators is the problem of classifying endomorphisms (or at least certain classes of them) with respect to some equivalence or other. For linear operators in pure algebra one considers, as a rule, the general categorical equivalence of endomorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934068.png" />; it is called similarity. In other words, linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934070.png" /> acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934071.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934072.png" />, respectively, are similar if for some isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934073.png" /> the diagram

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934074.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>

commutes. Equivalence of continuous linear operators in (general) Banach spaces is called topological equivalence, and is understood in the general categorical sense, this time in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934075.png" />; this implies that the diagram (1) commutes for some topological isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934076.png" />. For linear operators in Hilbert spaces one chooses unitary equivalence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934077.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934078.png" /> as basis, corresponding to the requirement that the diagram (1) commutes for some unitary (see below) operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934079.png" />.

Besides this, in the theory of linear operators between spaces with a topology there are important problems of approximating various classes of linear operators by operators of a comparatively simple structure. A significant role is played by problems about finding the general form of linear operators on concrete spaces, most frequently function spaces.

Among the invariants of similarity the most important are the spectrum and the number of invariant subspaces of given dimension. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934080.png" /> be a linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934081.png" />. Its spectrum is the subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934082.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934083.png" /> consisting of those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934084.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934085.png" /> has no inverse. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934086.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934087.png" /> is said to be invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934088.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934089.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934090.png" />. Besides the kernel and the image of a linear operator, examples are the one-dimensional subspaces containing the so-called eigen vectors of the operator, that is, those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934092.png" />, for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934093.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934094.png" />. The element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934095.png" />, called an eigen value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934096.png" />, automatically belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934097.png" />'s spectrum.

The concept of a linear operator is a special case of the concept of a morphism of modules, which is obtained by replacing the field by an arbitrary ring. In many respects morphisms of modules are not like linear operators in their properties, but results about the latter form one of the stimuli for studying them.

==Linear operators on finite-dimensional spaces==
(without additional structure). The main analytic apparatus for such linear operators is the matrix notation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934098.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l05934099.png" /> be spaces with fixed bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340100.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340101.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340103.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340104.png" /> be a linear operator, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340105.png" /> be the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340106.png" />-th coefficients of the expansion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340107.png" /> with respect to the second basis. Then the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340108.png" />-dimensional matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340109.png" /> is called the matrix of the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340110.png" /> in the bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340112.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340113.png" /> the matrix notation is usually taken in coincident bases (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340115.png" />). By transfer to other bases the matrix of a linear operator is changed by simple formulas.

To various characteristics of linear operators correspond, as a rule, effectively computable characteristics of their matrices. For example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340117.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340118.png" /> is the [[Rank|rank]] of the matrix; in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340119.png" /> is an isomorphism if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340120.png" />; this condition is equivalent to the [[Determinant|determinant]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340121.png" /> being non-zero. To algebraic operations on linear operators correspond operations of the same name on their matrices, taken in fixed bases.

Linear operators are similar if and only if they can be described (each in its  "own"  basis) by the same matrix. The eigen values of a linear operator are the roots of the [[Characteristic polynomial|characteristic polynomial]] of its matrix. This implies that every linear operator on a finite-dimensional space over an algebraically closed field (for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340122.png" />) has at least one eigen vector. The spectrum of a linear operator on a finite-dimensional space over an arbitrary field is the set of its eigen values.

The problem of classifying endomorphisms of finite-dimensional spaces over an algebraically closed field has been completely solved, and the similarity classes have been described in terms of the invariant subspaces. The main step here is the following classification theorem, discovered by C. Jordan (who considered the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340123.png" />). A matrix with a fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340124.png" /> on the main diagonal, ones immediately above it and zeros elsewhere, is called a Jordan block, and a block matrix with Jordan blocks (of different dimensions and with different <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340125.png" />) on the main diagonal and zeros elsewhere is called a Jordan matrix. Then every endomorphism can be written in some basis as a Jordan matrix. The search for such a basis is called reduction of a linear operator to Jordan form. Since linear operators described by Jordan matrices are similar if and only if their matrices coincide up to a permutation of the Jordan blocks, the theorem means that Jordan matrices present, after suitable identification, a complete set of similarity invariants.

==Continuous linear operators on Banach spaces.==
The foundations of this theory were laid by S. Banach (see [[#References|[4]]]).

===Examples.===


1) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340126.png" /> (from now on, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340127.png" />): the linear operator of multiplication by a bounded sequence of numbers; the linear operator of left (respectively, right) shift, which takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340128.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340129.png" /> (respectively, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340130.png" />).

2) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340131.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340132.png" />: the linear operator of multiplication by a continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340133.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340134.png" />; the linear operator of indefinite integration, which takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340135.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340136.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340137.png" /></td> </tr></table>

3) In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340138.png" />: the linear operator of a shift by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340139.png" />, which takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340140.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340141.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340142.png" />.

4) From <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340143.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340144.png" />: The  "classical"  Fourier operator, which takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340145.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340146.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340147.png" /></td> </tr></table>

A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340148.png" /> between Banach spaces is continuous if and only if it is bounded, that is, the image of every bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340149.png" /> is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340150.png" />, or equivalently, if there is a (finite) number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340151.png" />, called the operator norm (a similar assertion is also true for arbitrary normed spaces). The continuous linear operators from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340152.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340153.png" /> form a subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340154.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340155.png" /> which is a Banach space with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340156.png" />. The subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340157.png" />, often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340158.png" />, is even a Banach algebra with respect to operator multiplication. The class of these algebras is universal in the sense that every Banach algebra is topologically isomorphic to a subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340159.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340160.png" />. The subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340161.png" /> consisting of topological isomorphisms is open and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340162.png" /> contains the open ball of unit radius with centre at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340163.png" />.

A fundamental role in the  "Banach"  theory of linear operators is played by two theorems, which together with the [[Hahn–Banach theorem|Hahn–Banach theorem]] (see also [[Linear functional|Linear functional]]) are called the three fundamental principles of linear analysis. The simplest form of the  "principle of uniform boundedness"  is the Banach–Steinhaus theorem: If continuous linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340165.png" /> are such that for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340166.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340167.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340168.png" />. The simplest form of the  "open mapping principle"  is Banach's theorem: If a continuous linear operator has an inverse, then this inverse operator is automatically continuous. The requirement that the spaces be  "Banach"  (complete) spaces is essential in both theorems.

The spectrum of a continuous linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340169.png" />, usually considered for complex spaces, is a non-empty compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340170.png" />. In the case of an infinite-dimensional space the set of eigen values of a linear operator, called the point spectrum, generally forms only part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340171.png" />. For the remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340172.png" /> one always has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340173.png" />; these numbers split into two sets, the continuous spectrum and the residual spectrum, according as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340174.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340175.png" /> or not. For example, the spectrum of the linear operator of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340176.png" /> is the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340177.png" />, but in the case of spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340178.png" /> all its points belong to the continuous spectrum, and in the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340179.png" /> they belong to the residual spectrum.

The operator-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340180.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340181.png" /> is called the resolvent of the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340182.png" />. It is useful, particularly because for every function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340183.png" />, holomorphic in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340184.png" /> of the spectrum, it makes it possible to consider the linear operator (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340185.png" />)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340186.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340187.png" /> is a smooth contour in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340188.png" /> that bounds the spectrum. In this way one can construct a  "holomorphic operator calculus" ; as for  "holomorphic multi-operator calculus" , that is, loosely speaking, giving a reasonable sense to the concept of a holomorphic function of several  "operator"  variables, the problem of constructing it turned out to be far more difficult. Results that are in a certain sense close to final have been obtained by methods of homological algebra [[#References|[5]]].

In the theory of linear operators between Banach spaces an important operation is that of going over from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340189.png" /> to its so-called adjoint linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340190.png" /> (cf. also [[Adjoint operator|Adjoint operator]]), which is defined by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340191.png" /></td> </tr></table>

This operation has the properties

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340192.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340193.png" /></td> </tr></table>

(on the assumption that the left-hand sides of the equalities make sense). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340194.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340195.png" /> are reflexive (cf. [[Reflexive space|Reflexive space]]), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340196.png" />.

Some of the most important classes of linear operators in Banach spaces are the following.

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340197.png" /> is called a [[Compact operator|compact operator]], or a [[Completely-continuous operator|completely-continuous operator]], if it maps any bounded set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340198.png" /> to a totally-bounded (that is, pre-compact) set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340199.png" />. Every linear operator that can be approximated in the operator norm by finite-dimensional (having finite-dimensional image) linear operators is compact. For the overwhelming majority of classical Banach spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340200.png" /> the converse is also true: Every compact linear operator from any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340201.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340202.png" /> can be approximated by finite-dimensional operators. The so-called approximation problem (is this true for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340203.png" />?) has been negatively answered [[#References|[6]]].

The identity linear operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340204.png" /> is compact if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340205.png" /> is finite dimensional (Riesz' theorem). The spectrum of a compact linear operator is a set containing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340206.png" /> that is at most countable. If the spectrum is infinite, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340207.png" /> is the unique limit point of it. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340208.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340209.png" />, is an eigen value, and the eigen vectors corresponding to it form a finite-dimensional space.

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340210.png" /> is called a [[Nuclear operator|nuclear operator]] if it can be represented as an absolutely convergent series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340211.png" /> consisting of one-dimensional linear operators; such an operator is necessarily compact. By means of nuclear linear operators one can define a class of topological vector spaces, called nuclear spaces, which is important in analysis (cf. [[Nuclear space|Nuclear space]]).

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340212.png" /> is called a [[Fredholm-operator(2)|Fredholm operator]] if its kernel and cokernel are finite dimensional; the image of such a linear operator is necessarily closed. The main characteristic of a linear Fredholm operator is its index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340213.png" />. The set of linear Fredholm operators is open in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340214.png" /> and the index on it (in contrast to the dimensions of the kernel and cokernel taken separately) is a locally constant function. The forerunner of the theory of linear Fredholm operators was the theory of integral equations. Set up by E. Fredholm, who essentially proved (and D. Hilbert perceived this  "geometrical"  background) that a linear operator of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340215.png" /> with compact <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340216.png" /> is a Fredholm operator and has index zero (see [[Fredholm alternative|Fredholm alternative]]). Important specific classes of linear Fredholm operators arise in considering elliptic differential expressions on manifolds. The problem of calculating their index required the apparatus of algebraic topology [[#References|[8]]] (cf. also [[Index formulas|Index formulas]]).

The problem of classifying endomorphisms of Banach spaces, in view of Jordan's theorem, which gives a finite-dimensional model of its solution, can be treated  "to a first approximation"  as the problem of constructing an appropriate infinite-dimensional analogue of the Jordan form of a linear operator. However, it is far from being solved: to say nothing about a  "sufficient"  set of invariant subspaces, it is not known so far (1989) whether every operator that acts on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340217.png" /> has at least one non-trivial (distinct from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340218.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340219.png" />) invariant subspace. If a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340220.png" /> that acts on an arbitrary Banach space is not proportional to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340221.png" /> and commutes with some compact linear operator, then the set of all linear operators that commute with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340222.png" /> is a non-trivial invariant subspace (see [[#References|[9]]]).

Apart from the topology of a normed space, there are topologies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340223.png" /> that specify the structure of a [[Locally convex space|locally convex space]] in it. The most important are the strong and weak operator topologies, specified, respectively, by the systems of semi-norms

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340224.png" /></td> </tr></table>

and

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340225.png" /></td> </tr></table>

The theory of continuous linear operators on topological vector spaces (an example of such a linear operator is the Fourier transform on generalized functions) was developed on the basis of the  "Banach"  theory. A significant place in it is taken by the investigation of the possibility of generalizing Banach's classical theorems and others; it led to the introduction of a number of important classes of spaces. For example, the theorem on the equivalence of continuity and boundedness, the Banach–Steinhaus theorem and Banach's theorem on the inverse operator are not true for arbitrary, at least locally convex, separable complete spaces. At the same time, the first of these theorems is true if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340226.png" /> is a bornological space, the second if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340227.png" /> is a [[Barrelled space|barrelled space]], and the third if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340228.png" /> is absolutely complete and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340229.png" /> is a barrelled space.

==Continuous linear operators on Hilbert spaces==
(finite-dimensional and infinite-dimensional). Their theory was first formalized in the work of Hilbert [[#References|[10]]] on integral equations and infinite quadratic forms.

===Examples.===


1) All examples of linear operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340230.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340231.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340232.png" /> considered above, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340233.png" />.

2) The integral operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340234.png" /> that takes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340235.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340236.png" />,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340237.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340238.png" /> is a square-integrable function on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340239.png" />. Such a linear operator is always compact.

3) The Fourier operator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340240.png" /> is uniquely defined by the fact that it coincides with the classical Fourier operator (see above) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340241.png" />.

The concepts and facts are given below in the form that they have for spaces over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340242.png" />: namely, in complex spaces the theory has turned out to be most important and meaningful.

The special position of the  "Hilbert"  theory of linear operators in the background of the  "Banach"  theory is determined by the sharply increasing role of the concept of the [[Adjoint operator|adjoint operator]]. Since a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340243.png" /> and its adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340244.png" /> are anti-isomorphic, the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340245.png" /> is naturally identified with the linear operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340246.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340247.png" /> uniquely defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340248.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340249.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340250.png" />; under this identification the linear operator adjoint to an endomorphism in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340251.png" /> again acts in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340252.png" />. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340253.png" /> there arises an important additional structure — the operation of transition from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340254.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340255.png" />, which has involutory properties and with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340256.png" /> is a [[C*-algebra|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340257.png" />-algebra]]. In fact, every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340258.png" />-algebra is isometrically <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340259.png" />-isomorphic to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340260.png" />-subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340261.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340262.png" /> (the Gel'fand–Naimark theorem).

The following classes of linear operators are characteristic of Hilbert spaces.

A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340263.png" /> is called a [[Self-adjoint operator|self-adjoint operator]], or a [[Hermitian operator|Hermitian operator]], if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340264.png" />. A self-adjoint linear operator equal to its square is called a projector (projection operator); such a linear operator can be realized as the operator of orthogonal projection onto a closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340265.png" />. A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340266.png" /> is called a [[Unitary operator|unitary operator]] (in the case of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340267.png" />, an orthogonal operator) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340268.png" />, or, equivalently, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340269.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340270.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340271.png" />. A linear operator is unitary if and only if it is an isomorphism that preserves norms. Self-adjoint and unitary endomorphisms are special cases of a [[Normal operator|normal operator]]: A linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340272.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340273.png" />.

The linear operator of multiplication by a sequence (respectively, a function) in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340274.png" /> (respectively, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340275.png" />) is self-adjoint if and only if this sequence (function) is real valued. A linear integral operator is self-adjoint if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340276.png" /> almost-everywhere. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340277.png" /> shift operators and the Fourier operator are unitary. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340278.png" /> the linear operators of left and right shift are adjoint to one another and are not normal. The spectrum of a self-adjoint linear operator lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340279.png" />, and the spectrum of a unitary operator lies on the unit circle in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340280.png" />. The spectrum of a projector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340281.png" /> consists of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340282.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340283.png" />. The spectrum of a normal linear operator can be any compact set in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340284.png" />. The eigen vectors of a self-adjoint linear operator corresponding to different eigen values are orthogonal.

The classes of linear operators described above are used in a large number of branches of mathematics and physics, including quantum mechanics (where self-adjoint linear operators are interpreted as observables), the theory of representations and harmonic analysis, the theory of differential equations, and the theory of dynamical systems.

All information about a linear operator acting in a finite-dimensional Hilbert space is provided by its matrix notation in an orthogonal basis. In this notation transition to the adjoint linear operator corresponds to taking the matrix that is the complex conjugate of the transposed matrix; as a consequence, for the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340285.png" /> of a self-adjoint linear operator one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340286.png" />. The classical theorem  "on reduction to diagonal form"  asserts that every normal (in the case of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340287.png" />, every self-adjoint) linear operator can be written in some orthonormal basis as a diagonal matrix. Together with the fact that linear operators are unitarily equivalent if and only if they can be described (each in its own orthogonal basis) by the same matrices, this theorem implies that the sets of eigen values (taking account of their multiplicity) form a complete system of invariants of unitary equivalence for normal linear operators.

The theorem on reduction to diagonal form turned out to be  "happier"  than Jordan's theorem in that a meaningful infinite-dimensional analogue was found for it, the spectral theorem for normal linear operators, discovered in 1912: A normal linear operator can be represented uniquely as an operator-valued Stieltjes integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340288.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340289.png" /> is a countably-additive regular (in the sense of the strong operator topology, for example) function on the Borel subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340290.png" /> that takes values among projectors and is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340291.png" /> (see [[Spectral measure|Spectral measure]]). A consequence is that every normal linear operator can be approximated by linear combinations of projectors.

On the basis of Hilbert's theorem a complete classification up to unitary equivalence of normal linear operators (in a space of arbitrary dimension) has been obtained [[#References|[11]]]; in particular, it turns out that every normal linear operator is unitarily equivalent to the operator of multiplication by a bounded measurable function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340292.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340293.png" /> is a measurable space with finite measure. For compact normal linear operators in a separable space the picture is simplified: such a linear operator has as orthonormal basis its own eigen vectors and is therefore unitarily equivalent to the operator of multiplication by a sequence acting in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340294.png" /> (the sequence necessarily converges to zero).

In the case of a normal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340295.png" /> the spectral theorem makes it possible to assign a meaning to the function expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340296.png" /> for a wider class of functions on the spectrum than holomorphic functions, for example, for a continuous <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340297.png" /> the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340298.png" /> is defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340299.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340300.png" /> is self-adjoint, then the linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340301.png" /> is unitary.

The main effort has been concentrated on the study of different classes of non-normal linear operators, in the first place abstract Volterra operators, condensing operators and spectral operators (see [[#References|[11]]]–[[#References|[13]]], and [[Volterra operator|Volterra operator]]; [[Spectral operator|Spectral operator]]; [[Condensing operator|Condensing operator]]). Although regularities comparable with the regularity in the spectral theorem have still (1989) not been discovered, nevertheless deep results have been obtained in this direction.

The development of analysis and its applications, in the first place differential equations and quantum mechanics, forces one to go beyond the class of bounded (that is, continuous) linear operators. Strictly speaking, an [[Unbounded operator|unbounded operator]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340302.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340303.png" /> is not a linear operator in the sense accepted here, since it is defined not on the whole of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340304.png" />, but only on a dense subspace of it, as a rule. Typical examples are the linear operator of multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340305.png" /> and differentiation in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l059/l059340/l059340306.png" />. For unbounded linear operators analogues of the spectrum, the adjoint linear operator and the classes of linear operators considered above have been defined. Although their theory is more complicated than the theory of bounded linear operators, a number of deep results of the latter has been generalized meaningfully. The most important of these is an analogue of the spectral theorem, discovered by J. von Neumann for unbounded self-adjoint operators.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Peano,   "Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann" , Bocca  (1888)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,   "Lectures on linear algebra" , Interscience  (1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Toeplitz,   "Ueber die Auflösung unendlichvieler linearer Gleichungen mit unendlichvielen Unbekannten"  ''Rend. Circolo Mat. Palermo'' , '''28'''  (1909)  pp. 88–96</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.S. Banach,   "A course of functional analysis" , Kiev  (1948)  (In Ukrainian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.L. Taylor,   "A general framework for a multi-operator functional calculus"  ''Advances in Math.'' , '''9''' :  2  (1972)  pp. 183–252</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Enflo,   "A counterexample to the approximation problem in Banach spaces"  ''Acta Math.'' , '''130'''  (1973)  pp. 309–317</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H.H. Schaefer,   "Topological vector spaces" , Macmillan  (1966)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> , ''Functional analysis'' , Math. Reference Library , Moscow  (1972)  (In Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  V.I. Lomonosov,   "Invariant subspaces of a family of operators that commute with a completely continuous operator"  ''Funktsional. Anal. i Prilozhen.'' , '''3'''  (1973)  pp. 55–56  (In Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  D. Hilbert,   "Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen" , Chelsea, reprint  (1953)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. General theory" , '''1''' , Interscience  (1958)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral operators" , '''3''' , Interscience  (1971)</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,   M.G. Krein,   "Theory and applications of Volterra operators in Hilbert space" , Amer. Math. Soc.  (1970)  (Translated from Russian)</TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top">  B. Szökefalvi-Nagy,   Ch. Foiaş,   "Harmonic analysis of operators on Hilbert space" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top">  N. Bourbaki,   "Elements of mathematics. Algebra: Modules. Rings. Forms" , '''2''' , Addison-Wesley  (1975)  pp. Chapt.4;5;6  (Translated from French)</TD></TR></table>



====Comments====
For a fairly recent text on multi-dimensional functional calculus cf. [[#References|[a6]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  I.C. Gohberg,   S. Goldberg,   "Basic operator theory" , Birkhäuser  (1977)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I.C. [I.Ts. Gokhberg] Gohberg,   M.G. Krein,   "Introduction to the theory of linear nonselfadjoint operators" , ''Transl. Math. Monogr.'' , '''18''' , Amer. Math. Soc.  (1969)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P.R. Halmos,   "Finite-dimensional vector spaces" , v. Nostrand  (1958)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Pazy,   "Semigroups of linear operators and applications to partial differential equations" , Springer  (1983)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C. Pearcy (ed.) , ''Topics in operator theory'' , Amer. Math. Soc.  (1974)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  F.H. Vasilescu,   "Analytic functional calculus and spectral decompositions" , Reidel  (1982)</TD></TR></table>