View source for Extension of an operator

A [[Linear operator|linear operator]] whose graph contains the graph of the given linear operator. When the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370401.png" /> is an extension of a given operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370402.png" />, one writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370403.png" />. The usual problems in the theory of extensions are as follows: extend an operator maximally while preserving a specific property, or study extensions of an operator having various additional properties.

For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370404.png" /> be a given isometric operator on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370405.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370406.png" /> and range of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370407.png" />; then the isometric extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370408.png" /> are in one-to-one correspondence with the isometric mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e0370409.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704010.png" />. In particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704011.png" /> has unitary extensions if the dimensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704013.png" /> coincide.

==Extensions of symmetric operators.==
The most studied (and most important in applications) is the theory of self-adjoint extensions of symmetric operators on a Hilbert space. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704014.png" /> is symmetric if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704015.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704016.png" /> is the operator adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704017.png" />. Thus, the domain of definition of any symmetric extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704018.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704019.png" />, and these extensions are restrictions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704020.png" />. This reduces the description of the symmetric extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704021.png" /> to determining their domains of definition. A subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704022.png" /> is the domain of definition of some symmetric extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704023.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704024.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704025.png" />. It turns out that

<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/e/e037/e037040/e03704026.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704027.png" /> are deficiency subspaces (defect subspaces) (their dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704028.png" /> are called deficiency numbers (defect numbers)), and the symmetric extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704029.png" /> are in one-to-one correspondence with the isometric mappings from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704030.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704031.png" />: To any such mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704032.png" /> corresponds an extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704033.png" /> with domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704034.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704035.png" /> is the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704036.png" />. Self-adjoint extensions correspond to unitary operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704037.png" />, and hence exist if and only if the defect numbers are equal.

The domains of definition of extensions of symmetric operators can be conveniently described with the help of so-called (abstract) boundary conditions. Any linear functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704038.png" />, continuous relative to the norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704039.png" />, and equal to zero on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704040.png" />, is called a boundary value for the symmetric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704041.png" />; an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704042.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704043.png" /> is a boundary value, is called a boundary condition. Boundary values are determined by their values on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704044.png" />. If the defect numbers of a symmetric operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704045.png" /> are finite, then each of its symmetric extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704046.png" /> is determined by a family of boundary conditions, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704048.png" /> are boundary values. The families of boundary values determining self-adjoint extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704049.png" /> with defect numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704050.png" /> can be described as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704051.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704052.png" /> be orthonormal bases of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704054.png" />, respectively, and let, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704055.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/e/e037/e037040/e03704056.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/e/e037/e037040/e03704057.png" /></td> </tr></table>

Then any self-adjoint extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704058.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704059.png" /> is determined by the boundary conditions

<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/e/e037/e037040/e03704060.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704061.png" /> is a unitary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704062.png" />-matrix.

In some cases one can successfully prove the existence of self-adjoint extensions (and find some of them) without solving the difficult problem of determining the defect subspaces and defect numbers. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704063.png" /> commutes with an (anti-unitary) involution of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704064.png" />, then it has a self-adjoint extension. This is often used in the theory of differential operators, where the involution is taken to be complex conjugation in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704065.png" />. Equality of defect numbers also occurs in the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704066.png" /> has points of regular type on the real axis (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704067.png" /> is called a point of regular type if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704068.png" /> for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704069.png" /> and for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704070.png" />).

==Extensions of semi-bounded operators.==
An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704071.png" /> is called semi-bounded from below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704072.png" /> if its numerical range <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704073.png" /> lies in the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704074.png" />; an operator is called positive if it is semi-bounded from below by zero. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704075.png" /> is semi-bounded from below by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704076.png" />, then every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704077.png" /> is a point of regular type, the defect numbers are equal and self-adjoint extensions exist. One such an extension can be constructed as follows. The sesquilinear form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704078.png" /> defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704079.png" /> admits a closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704080.png" />. But, being a closed symmetric bilinear form, to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704081.png" /> corresponds a unique self-adjoint operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704082.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704083.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704084.png" /> is called the Friedrichs extension of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704085.png" /> and is semi-bounded. The greatest lower bound of its spectrum is equal to the greatest lower bound of the numerical range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704086.png" />. This is the unique self-adjoint extension whose domain of definition is contained in the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704087.png" />. Using the Friedrichs extension it is possible to describe other semi-bounded extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704088.png" /> (if the defect numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704089.png" /> are finite, then all its self-adjoint extensions are semi-bounded). For this it is sufficient to find all positive extensions of positive operators (the general case reduces to this by adding a multiple of the identity operator). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704090.png" /> be a positive operator and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704091.png" />. Then the positive self-adjoint extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704092.png" /> correspond uniquely to positive bounded operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704093.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704094.png" />; for every such operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704095.png" />, the subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704096.png" /> is the domain of definition of the corresponding extension (see ).

The construction of the Friedrichs extension can be generalized to the case of sectorial operators, i.e. operators whose numerical range is contained in some angle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704097.png" />. There is an extension, which is a maximal sectorial operator, whose numerical range lies in the same angle and which, like the Friedrichs extension, is minimal. The case of operators from a Banach space into its dual space has also been studied (see [[#References|[5]]]).

==Dissipative extensions.==
In some problems it is necessary to construct symmetric extensions of symmetric operators. A typical result is as follows. An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704098.png" /> is called dissipative if its numerical range lies in the left half-plane, and maximal dissipative if it is dissipative and does not have dissipative extensions. Every symmetric operator has an extension of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e03704099.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040100.png" /> is a maximal dissipative operator; all such extensions can be described by means of contractive mappings of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040101.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040102.png" /> (see [[#References|[8]]]).

==Extensions of differential operators.==
The theory of extensions of operators has important applications in the study of differential operators. Let

<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/e/e037/e037040/e037040103.png" /></td> </tr></table>

be a formal self-adjoint differential expression on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040104.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040105.png" /> be the subspace consisting of all functions with absolutely-continuous quasi-derivatives of orders <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040106.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040107.png" />-th quasi-derivative in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040108.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040109.png" /> be the subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040110.png" /> consisting of the functions whose supports do not contain the end points of the intervals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040111.png" /> be the operator given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040112.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040113.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040114.png" /> be its restriction to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040115.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040116.png" /> is symmetric, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040117.png" />; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040118.png" /> be its closure. In the regular case (that is, when the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040119.png" /> is finite and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040120.png" /> is summable), the domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040121.png" /> is formed by all functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040122.png" /> whose first <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040123.png" /> quasi-derivatives vanish at the end points of the interval. In the singular case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040124.png" /> is harder to describe (see [[#References|[2]]]). The defect numbers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040125.png" /> coincide and are equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040126.png" /> in the regular case and are at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040127.png" /> in the singular case. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040128.png" /> always has self-adjoint extensions; their spectra, spectral decompositions and resolvents are the basic objects of study in the theory of differential operators, because the choice of this or that self-adjoint extension is in fact the precise formulation of a certain spectral problem. This is particularly clear in the regular case, when the (abstract) boundary conditions, giving the domain of definition of a self-adjoint extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040129.png" />, are written in the form of ordinary boundary conditions:

<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/e/e037/e037040/e037040130.png" /></td> </tr></table>

for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040132.png" /> (this follows from the above description of (abstract) boundary conditions since, in the regular case, the boundary values are determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040133.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040134.png" />).

For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040135.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040136.png" /> is semi-bounded from below, and its Friedrichs extension corresponds to the boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040137.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040138.png" />.

In the general case, the self-adjoint extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040139.png" /> can be characterized as follows. Let

<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/e/e037/e037040/e037040140.png" /></td> </tr></table>

for all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040141.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040142.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040143.png" />. Then the limits

<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/e/e037/e037040/e037040144.png" /></td> </tr></table>

exist and satisfy

<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/e/e037/e037040/e037040145.png" /></td> </tr></table>

(Lagrange's formula). Thus, to describe the self-adjoint extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040146.png" /> it is sufficient to choose bases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040147.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040148.png" /> of the defect subspaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040149.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040150.png" /> (it is convenient to assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040151.png" />) and to put every unitary matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040152.png" /> into correspondence with the self-adjoint extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040153.png" /> whose domain of definition consists of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040154.png" /> that satisfy the boundary conditions

<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/e/e037/e037040/e037040155.png" /></td> </tr></table>

where

<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/e/e037/e037040/e037040156.png" /></td> </tr></table>

==Extensions corresponding to boundary value problems.==
Extensions of semi-bounded operators play a central role in the theory of elliptic boundary value problems. For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040157.png" /> be an elliptic differential expression of the second order in a region <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040158.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040159.png" />-dimensional space, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040160.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040161.png" /> be the minimal and maximal operators determined by this expression. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040162.png" /> is positive definite, its defect numbers are infinite and the defect subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040163.png" /> (called the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040165.png" />-harmonic functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040166.png" />) has a natural realization as a function space on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040167.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040168.png" />. Thus, various extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040169.png" /> correspond to various boundary conditions and thus determine various boundary value problems. In particular, the Friedrichs extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040170.png" /> is defined on all functions in the [[Sobolev space|Sobolev space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040171.png" /> that vanish on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040172.png" />, and the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040173.png" /> corresponds to the [[Dirichlet problem|Dirichlet problem]]:

<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/e/e037/e037040/e037040174.png" /></td> </tr></table>

The theory of partial differential equations gives rise to a number of general questions about extensions of symmetric operators, such as the question of uniqueness of a self-adjoint extension (so-called essential self-adjointness), whether operators that commute (in some sense) have commuting extensions, whether there exist intermediate extensions with given properties (for example, with conditions on the spectrum), etc. (see [[#References|[7]]]–[[#References|[9]]]).

==Extensions into an enlarged Hilbert space.==
Every symmetric operator acting on a Hilbert space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040175.png" /> can be extended to a self-adjoint operator acting on some space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040176.png" /> (see [[#References|[10]]]), and so every symmetric operator has a generalized spectral function. This is related to various results on extensions from a space and on dilatations (see [[#References|[11]]]). Thus, any contraction of a Hilbert space (that is, any operator of norm <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040177.png" />) can be extended to a co-isometric operator (that is, the adjoint of an isometric operator), and every contraction whose powers strongly converge to zero can be extended to a backwards one-sided shift (that is, the adjoint of a one-sided shift). Results about extensions into an enlarged space can be generalized to commutative families, semi-groups, etc.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Dunford,   J.T. Schwartz,   "Linear operators. Spectral theory" , '''2''' , Interscience  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M.A. Naimark,   "Lineare Differentialoperatoren" , Akademie Verlag  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  T. Kato,   "Perturbation theory for linear operators" , Springer  (1966)  pp. Chapt. X Sect. 3</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  M.G. Krein,   "Theory of selfadjoint extensions of semibounded Hermitian transformations and its applications I"  ''Mat. Sb.'' , '''20'''  (1947)  pp. 431–498  (In Russian)</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  M.G. Krein,   "Theory of selfadjoint extensions of semibounded Hermitian transformations and its applications II"  ''Mat. Sb.'' , '''21'''  (1947)  pp. 365–404  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  M.Sh. Birman,   "On the theory of selfadjoint extensions of positive definite operators"  ''Mat. Sb.'' , '''38'''  (1956)  pp. 431–450  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  R.S. Phillips,   "Dissipative operators and hyperbolic systems of partial differential equations"  ''Trans. Amer. Math. Soc.'' , '''90''' :  2  (1959)  pp. 193–254</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  K. Moren,   "Methods of Hilbert spaces" , PWN  (1967)  (Translated from Polish)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  Yu.M. [Yu.M. Berezanskii] Berezanskiy,   "Expansion in eigenfunctions of selfadjoint operators" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  S.G. Mikhlin,   "The problem of the minimum of a quadratic functional" , Holden-Day  (1965)  (Translated from Russian)</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M.A. Naimark,   "Spectral functions of a symmetric operator"  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''4'''  (1940)  pp. 277–318  (In Russian)</TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top">  B. Szökefalvi-Nagy,   Ch. Foiaş,   "Harmonic analysis of operators on Hilbert spaces" , North-Holland  (1970)  (Translated from French)</TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top">  L. Brown,   R. Douglas,   P. Fillmore,   "Unitary equivalence modulo the compact operators and extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040178.png" />-algebras" , ''Proc. Conf. Operator Theory'' , ''Lect. notes in math.'' , '''345''' , Springer  (1973)  pp. 58–128</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top">  W. Arveson,   "Notes on extensions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040179.png" />-algebras"  ''Duke Math. J.'' , '''44''' :  2  (1977)  pp. 329–355</TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top">  M. Reed,   B. Simon,   "Methods of contemporary mathematical physics" , '''II: Fourier analysis, self-adjointness''' , Acad. Press  (1975)</TD></TR></table>



====Comments====
The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040180.png" /> is another occasionally used notation for the (internal) direct sum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040181.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040182.png" /> signifies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040183.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040184.png" />.

When dealing with a differential expression

<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/e/e037/e037040/e037040185.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

it may happen that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040186.png" /> makes sense even though not all derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040188.png" />, exist; this is connected with possible non-differentiability of the coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040189.png" />. One therefore defines the quasi-derivatives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040190.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040191.png" /> corresponding to the expression (a1) above 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/e/e037/e037040/e037040192.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/e/e037/e037040/e037040193.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/e/e037/e037040/e037040194.png" /></td> </tr></table>

In addition one sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040195.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040196.png" />.

There exists another class of problems which go by the name of extension problems. These are matrix extension problems and operator extension problems, in which the matrices (operators) are incompletely given and it is required to  "fill-in"  the missing  "entries"  in such a way that the resulting matrix (operator) has certain specified properties. For instance, in one finite version of the Carathéodory–Toeplitz problem one has given a band around the diagonal of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040197.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040198.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040199.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040200.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040201.png" /> (and some other necessary conditions), and it is required to find the remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e037/e037040/e037040202.png" /> such that the resulting matrix has non-negative definite real part. There is a corresponding infinite version which is important in signal analysis.

Such problems relate to the [[Interpolation of operators|interpolation of operators]].