View source for Symbol of an operator

A scalar or matrix function associated with the operator and having properties that somehow reflect the properties of the operator. One usually supposes that the operators to which a symbol is assigned belong to an algebra. Then, as a rule, under summation of operators their symbols are added, and under multiplication they are multiplied, either up to terms which are small in some sense or exactly. The symbol of an operator usually takes values in an algebra (in particular, an operator algebra) that is simpler than the original one.

Usually symbols are associated with operators acting on function spaces. In this case a typical situation is that if an operator acts on functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915701.png" /> variables (or more generally, on functions on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915702.png" />-dimensional manifold), then its symbol is a function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915703.png" /> variables (or, on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915704.png" />-dimensional manifold). The theory of pseudo-differential operators (cf. [[Pseudo-differential operator|Pseudo-differential operator]]) has been constructed by using such symbols. The correspondence between symbols and operators is at the basis of quantization, under which the symbol is a classical observable and the operator itself is the corresponding quantum observable.

==Symbols of operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915705.png" />.==
Suppose one is given a polynomial

<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/s/s091/s091570/s0915706.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915707.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915708.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s0915709.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157011.png" /> are multi-indices (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157014.png" /> integers, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157015.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157016.png" />. Then it is possible to construct, by various methods, an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157017.png" /> acting on functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157018.png" /> by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157019.png" /> with the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157020.png" /> of multiplication by one of the coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157021.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157022.png" /> and by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157023.png" /> with the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157024.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157026.png" /> is an arbitrary constant (playing the role of Planck's constant). If the role of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157027.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157028.png" /> is changed, then different operators are obtained. If one puts

<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/s/s091/s091570/s09157029.png" /></td> </tr></table>

then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157030.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157032.png" />-symbol, or left symbol, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157033.png" />. The correspondence between left symbols and operators obtained in this way is a one-to-one correspondence between polynomials and differential operators with polynomial coefficients and may be extended to significantly broader classes of operators and symbols by using 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/s/s091/s091570/s09157034.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157035.png" />, for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157036.png" />-dimensional vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157039.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157040.png" />.

An operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157041.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157043.png" />-symbol, or right symbol, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157044.png" /> 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/s/s091/s091570/s09157045.png" /></td> </tr></table>

or, for more general symbols,

<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/s/s091/s091570/s09157046.png" /></td> </tr></table>

A more symmetric method for constructing an operator corresponding to a polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157047.png" /> is obtained if one introduces for non-commuting operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157049.png" /> the symmetrized product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157050.png" /> 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/s/s091/s091570/s09157051.png" /></td> </tr></table>

and then puts

<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/s/s091/s091570/s09157052.png" /></td> </tr></table>

Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157053.png" /> is called the Weyl symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157054.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157055.png" /> can be expressed in terms of its Weyl symbol 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/s/s091/s091570/s09157056.png" /></td> </tr></table>

A second quantization leads to the appearance of another two types of symbols of operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157057.png" />, Wick and anti-Wick. Namely, introduce the creation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157058.png" /> (cf. [[Creation operators|Creation operators]]) and the annihilation operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157059.png" /> (cf. [[Annihilation operators|Annihilation operators]]) and write a differential operator with polynomial coefficients in the form

<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/s/s091/s091570/s09157060.png" /></td> </tr></table>

or in the form

<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/s/s091/s091570/s09157061.png" /></td> </tr></table>

Then its Wick symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157062.png" /> and anti-Wick symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157063.png" /> are 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/s/s091/s091570/s09157064.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/s/s091/s091570/s09157065.png" /></td> </tr></table>

For formulas connecting the different types of symbols of an operator see [[#References|[1]]]–[[#References|[4]]].

==Symbols of operators on manifolds.==
Symbols of the type described above on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157066.png" /> are in one-to-one correspondence with operators of certain fairly broad classes, whereas on manifolds, as a rule, there are no natural symbols for which such a one-to-one correspondence exists. On manifolds an important role is played by the so-called principal symbol, which is defined for certain pseudo-differential operators and is a homogeneous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157067.png" />, the cotangent bundle of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157068.png" /> without the zero section. Its invertibility implies that the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157069.png" /> in question is elliptic and guarantees that a regularity theorem holds, that is, smoothness of the solutions of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157070.png" />, with a smooth right-hand side <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157071.png" />, and also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157072.png" /> is Fredholm (if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157073.png" /> is compact) in suitable Sobolev spaces. Under addition and multiplication of operators their principal symbols are added and multiplied, respectively. The principal symbol does not change when terms of lower order are added to the operator.

==Symbols of operators on manifolds with boundary.==
On a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157074.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157075.png" />, a pseudo-differential operator has the form of a matrix ([[#References|[5]]]–[[#References|[7]]]):

<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/s/s091/s091570/s09157076.png" /></td> </tr></table>

Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157078.png" /> are vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157079.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157081.png" /> are vector bundles over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157082.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157083.png" /> is a pseudo-differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157084.png" /> having the transmission property (cf. [[Transmission, condition of|Transmission, condition of]]); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157085.png" /> is a boundary operator, that is, an operator acting on certain boundary conditions (in general, pseudo-differential); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157086.png" /> is a coboundary operator, or operator of potential type; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157087.png" /> is a singular Green operator (i.e. a product of boundary and coboundary operators or a more general operation of a similar structure); and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157088.png" /> is a pseudo-differential operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157089.png" />. The operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157090.png" /> has symbols of two types: interior and boundary. The interior symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157091.png" /> is the ordinary symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157092.png" />, which is a function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157093.png" />, more precisely, it is a section of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157094.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157095.png" /> is the canonical projection. The boundary symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157096.png" /> is the function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157097.png" /> that takes values in the operators on the semi-axis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157098.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s09157099.png" /> by freezing the coefficients of the principal part at points of the boundary (in coordinates in which the boundary is a hyperplane) followed by Fourier transformation with respect to the tangential variables. The invertibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570100.png" /> is the usual ellipticity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570101.png" /> (cf. [[Elliptic operator|Elliptic operator]]). If this ellipticity is assumed, then the invertibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570102.png" /> in classes of decreasing functions on the semi-axis is in fact the ellipticity condition of the boundary problem defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570103.png" />, or the so-called Shapiro–Lopatinskii condition. Thus it is natural to call the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570104.png" /> the symbol of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570105.png" />. If both symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570106.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570107.png" /> are invertible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570108.png" /> is called elliptic and, in this case, the usual theorems on regularity and being Fredholm (the latter when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091570/s091570109.png" /> is compact) are true.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F.A. Berezin,   "The method of second quantization" , Acad. Press  (1966)  (Translated from Russian)  (Revised (augmented) second edition: Kluwer, 1989)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.P. Maslov,   "Operator methods" , MIR  (1976)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M.A. Shubin,   "Pseudo-differential operators and spectral theory" , Springer  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  F.A. Berezin,   "Wick and anti-Wick operator symbols"  ''Math. USSR Sb.'' , '''15''' :  4  (1970)  pp. 577–606  ''Mat. Sb.'' , '''86''' :  4  (1971)  pp. 578–610</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L. Boutet de Monvel,   "Boundary problems for pseudo-differential operators"  ''Acta Math.'' , '''126'''  (1971)  pp. 11–51</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  S. Rempel,   B.-W. Schulze,   "Index theory of elliptic boundary problems" , Akademie Verlag  (1982)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  G. Grubb,   "Functional calculus of pseudo-differential boundary problems" , Birkhäuser  (1986)</TD></TR></table>



====Comments====
The notion of a symbol also plays an important role in the theory of singular integral operators (one-dimensional as well as multi-dimensional ones, cf. [[Singular integral equation|Singular integral equation]]). The investigation of systems of equations involving such operators has led to the notion of a matrix symbol. The latter notion is not only important for systems; it appears also in the study of singular integral operators with piecewise-continuous scalar coefficients.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  N.Ya. Krupnik,   "Banach algebras with symbol and singular integral operators" , Birkhäuser  (1987)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S.G. Mikhlin,   "Singular integral operators" , Akademie Verlag  (1986)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L. Boutet de Monvel,   "A course on pseudodifferential operators and their applications" , Math. Dept. Duke University  (1976)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.E. Taylor,   "Pseudo-differential operators" , Princeton Univ. Press  (1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''3''' , Springer  (1985)  pp. Chapts. 23–24</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B.A. Plamenevskii,   "Algebras of pseudodifferential operators" , Kluwer  (1989)  (Translated from Russian)</TD></TR></table>