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 $ n $ variables (or more generally, on functions on an $ n $- dimensional manifold), then its symbol is a function of $ 2n $ variables (or, on a $ 2n $- dimensional manifold). The theory of pseudo-differential operators (cf. 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 $ \mathbf R ^ {n} $.

Suppose one is given a polynomial

$$ a ( p, q) = \ \sum _ {| \alpha + \beta | \leq m } a _ {\alpha \beta } q ^ \alpha p ^ \beta , $$

where $ q, p \in \mathbf R ^ {n} $, $ q = ( q _ {1} \dots q _ {n} ) $, $ p = ( p _ {1} \dots p _ {n} ) $, $ \alpha $ and $ \beta $ are multi-indices (that is, $ \alpha = ( \alpha _ {1} \dots \alpha _ {n} ) $, $ \alpha _ {i} \geq 0 $, $ \alpha _ {i} $ integers, $ | \alpha | = \alpha _ {1} + \dots + \alpha _ {n} $), $ a _ {\alpha \beta } \in \mathbf C $. Then it is possible to construct, by various methods, an operator $ A $ acting on functions on $ \mathbf R ^ {n} $ by replacing $ q _ {j} $ with the operator $ \widehat{q} _ {j} $ of multiplication by one of the coordinates $ x _ {j} $ in $ \mathbf R ^ {n} $ and by replacing $ p _ {j} $ with the operator $ \widehat{p} _ {j} = ( h/i) ( \partial / \partial x _ {j} ) $, where $ i = \sqrt - 1 $ and $ h $ is an arbitrary constant (playing the role of Planck's constant). If the role of $ p $ and $ q $ is changed, then different operators are obtained. If one puts

$$ A = a ( {\widehat{q} } ^ { 2 } , {\widehat{p} } ^ { 1 } ) = \ \sum _ {| \alpha + \beta | \leq m } a _ {\alpha \beta } \widehat{q} {} ^ \alpha \widehat{p} {} ^ \beta , $$

then $ a ( q, p) $ is called the $ qp $- symbol, or left symbol, of $ A $. 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

$$ ( Au) ( x) = \ { \frac{1}{( 2 \pi h) ^ {n} } } \int\limits e ^ {i ( x - y) \xi /h } a ( x, \xi ) u ( y) dy d \xi , $$

where $ z \xi = z _ {1} \xi _ {1} + \dots + z _ {n} \xi _ {n} $, for $ n $- dimensional vectors $ z = ( z _ {1} \dots z _ {n} ) $ and $ \xi = ( \xi _ {1} \dots \xi _ {n} ) $, $ dy = dy _ {1} \dots dy _ {n} $, $ d \xi = d \xi _ {1} \dots d \xi _ {n} $.

An operator $ A $ with $ pq $- symbol, or right symbol, $ a ( q, p) $ is defined by the formula

$$ A = a ( {\widehat{q} } ^ { 1 } , {\widehat{p} } ^ { 2 } ) = \ \sum _ {| \alpha + \beta | \leq m } a _ {\alpha \beta } \widehat{p} {} ^ \beta \widehat{q} {} ^ \alpha $$

or, for more general symbols,

$$ ( Au) ( x) = \ \frac{1}{( 2 \pi h) ^ {n} } \int\limits e ^ {i ( x - y) \xi /h } a ( y, \xi ) u ( y) dy d \xi . $$

A more symmetric method for constructing an operator corresponding to a polynomial $ a ( q, p) $ is obtained if one introduces for non-commuting operators $ B $, $ C $ the symmetrized product $ ( B ^ {k} C ^ {l} ) $ by the formula

$$ ( sB + tC) ^ {n} = \ \sum _ {k + l = n } \frac{n! }{k! l! } s ^ {k} t ^ {l} ( B ^ {k} C ^ {l} ), $$

and then puts

$$ A = \sum _ {| \alpha + \beta | \leq m } a _ {\alpha \beta } ( \widehat{q} {} _ {1} ^ {\alpha _ {1} } \widehat{p} {} _ {1} ^ {\beta _ {1} } ) \dots ( \widehat{q} {} _ {n} ^ {\alpha _ {n} } \widehat{p} {} _ {n} ^ {\beta _ {n} } ). $$

Then $ a ( q, p) $ is called the Weyl symbol of $ A $. The operator $ A $ can be expressed in terms of its Weyl symbol by the formula

$$ ( Au) ( x) = \ \frac{1}{( 2 \pi h) ^ {n} } \int\limits e ^ {i ( x - y) \xi /h } a \left ( { \frac{x + y }{2} } , \xi \right ) u ( y) dy d \xi . $$

A second quantization leads to the appearance of another two types of symbols of operators on $ \mathbf R ^ {n} $, Wick and anti-Wick. Namely, introduce the creation operator $ a _ {j} ^ {+} = \widehat{q} _ {j} - i \widehat{p} _ {j} $( cf. Creation operators) and the annihilation operator $ a _ {j} ^ {-} = \widehat{q} _ {j} + i \widehat{p} _ {j} $( cf. Annihilation operators) and write a differential operator with polynomial coefficients in the form

$$ A = \ \sum _ {\alpha , \beta } c _ {\alpha \beta } ( a ^ {+} ) ^ \alpha a ^ \beta $$

or in the form

$$ A = \ \sum _ {\alpha , \beta } c _ {\alpha \beta } ^ \prime a ^ \beta ( a ^ {+} ) ^ \alpha . $$

Then its Wick symbol $ c ( q, p) $ and anti-Wick symbol $ a ( q, p) $ are given by the formulas

$$ c ( q, p) = \ \sum _ {\alpha , \beta } c _ {\alpha \beta } ( q - ip) ^ \alpha ( q + ip) ^ \beta , $$

$$ a ( q, p) = \sum _ {\alpha , \beta } c _ {\alpha \beta } ^ \prime ( q - ip) ^ \alpha ( q + ip) ^ \beta . $$

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

Symbols of operators on manifolds.

Symbols of the type described above on $ \mathbf R ^ {n} $ 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 $ T ^ {*} X \setminus 0 $, the cotangent bundle of the manifold $ X $ without the zero section. Its invertibility implies that the operator $ A $ in question is elliptic and guarantees that a regularity theorem holds, that is, smoothness of the solutions of the equation $ Au = f $, with a smooth right-hand side $ f $, and also that $ A $ is Fredholm (if $ X $ 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 $ X $ with boundary $ Y $, a pseudo-differential operator has the form of a matrix ([5]–[7]):

$$ \mathfrak A = \ \left \| \begin{array}{cc} A + B & K \\ T & Q \\ \end{array} \right \| : \ \begin{array}{ccc} \Gamma ( E _ {1} ) &{} &\Gamma ( E _ {2} ) \\ \oplus & \rightarrow &\oplus \\ \Gamma ( G _ {1} ) &{} &\Gamma ( G _ {2} ) \\ \end{array} . $$

Here $ E _ {1} $, $ E _ {2} $ are vector bundles over $ X $; $ G _ {1} $, $ G _ {2} $ are vector bundles over $ Y $; $ A $ is a pseudo-differential operator on $ X $ having the transmission property (cf. Transmission, condition of); $ T $ is a boundary operator, that is, an operator acting on certain boundary conditions (in general, pseudo-differential); $ K $ is a coboundary operator, or operator of potential type; $ B $ is a singular Green operator (i.e. a product of boundary and coboundary operators or a more general operation of a similar structure); and $ Q $ is a pseudo-differential operator on $ Y $. The operator $ \mathfrak A $ has symbols of two types: interior and boundary. The interior symbol $ \sigma ^ {0} ( \mathfrak A ) $ is the ordinary symbol of $ A $, which is a function on $ T ^ {*} X \setminus 0 $, more precisely, it is a section of the bundle $ \mathop{\rm Hom} ( \pi ^ {*} E _ {1} , \pi ^ {*} E _ {2} ) $, where $ \pi : T ^ {*} X \setminus 0 \rightarrow X $ is the canonical projection. The boundary symbol $ \sigma _ {Y} ( \mathfrak A ) $ is the function on $ T ^ {*} Y \setminus 0 $ that takes values in the operators on the semi-axis $ [ 0, \infty ) $ obtained from $ \mathfrak A $ 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 $ \sigma ^ {0} ( \mathfrak A ) $ is the usual ellipticity of $ A $( cf. Elliptic operator). If this ellipticity is assumed, then the invertibility of $ \sigma _ {Y} ( \mathfrak A ) $ in classes of decreasing functions on the semi-axis is in fact the ellipticity condition of the boundary problem defined by $ \mathfrak A $, or the so-called Shapiro–Lopatinskii condition. Thus it is natural to call the pair $ ( \sigma ^ {0} ( \mathfrak A ), \sigma _ {Y} ( \mathfrak A )) $ the symbol of $ \mathfrak A $. If both symbols $ \sigma ^ {0} ( \mathfrak A ) $ and $ \sigma _ {Y} ( \mathfrak A ) $ are invertible, then $ \mathfrak A $ is called elliptic and, in this case, the usual theorems on regularity and being Fredholm (the latter when $ X $ is compact) are true.

References

[1]	F.A. Berezin, "The method of second quantization" , Acad. Press (1966) (Translated from Russian) (Revised (augmented) second edition: Kluwer, 1989)
[2]	V.P. Maslov, "Operator methods" , MIR (1976) (Translated from Russian)
[3]	M.A. Shubin, "Pseudo-differential operators and spectral theory" , Springer (1987) (Translated from Russian)
[4]	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
[5]	L. Boutet de Monvel, "Boundary problems for pseudo-differential operators" Acta Math. , 126 (1971) pp. 11–51
[6]	S. Rempel, B.-W. Schulze, "Index theory of elliptic boundary problems" , Akademie Verlag (1982)
[7]	G. Grubb, "Functional calculus of pseudo-differential boundary problems" , Birkhäuser (1986)

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). 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

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