Functional calculus

Theory allowing one to apply mathematical functions to mathematical operators

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In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.)

If $f$ {\displaystyle f} is a function, say a numerical function of a real number, and $M$ {\displaystyle M} is an operator, there is no particular reason why the expression $f(M)$ {\displaystyle f(M)} should make sense. If it does, then we are no longer using $f$ {\displaystyle f} on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of $f(x)=x^{2}$ {\displaystyle f(x)=x^{2}} and $M$ {\displaystyle M} an $n\times n$ {\displaystyle n\times n} matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation.

The most immediate case is to apply polynomial functions to a square matrix, extending what has just been discussed. In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator $T$ {\displaystyle T}. This family is an ideal in the ring of polynomials. Furthermore, it is a nontrivial ideal: let $N$ {\displaystyle N} be the finite dimension of the algebra of matrices, then $\{I,T,T^{2},\ldots ,T^{N}\}$ {\displaystyle \{I,T,T^{2},\ldots ,T^{N}\}} is linearly dependent. So $\sum _{i=0}^{N}\alpha _{i}T^{i}=0$ {\displaystyle \sum _{i=0}^{N}\alpha _{i}T^{i}=0} for some scalars $\alpha _{i}$ {\displaystyle \alpha _{i}}, not all equal to 0. This implies that the polynomial $\sum _{i=0}^{N}\alpha _{i}x^{i}$ {\displaystyle \sum _{i=0}^{N}\alpha _{i}x^{i}} lies in the ideal. Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial $m$ {\displaystyle m}. Multiplying by a unit if necessary, we can choose $m$ {\displaystyle m} to be monic. When this is done, the polynomial $m$ {\displaystyle m} is precisely the minimal polynomial of $T$ {\displaystyle T}. This polynomial gives deep information about $T$ {\displaystyle T}. For instance, a scalar $\alpha$ {\displaystyle \alpha } is an eigenvalue of $T$ {\displaystyle T} if and only if $\alpha$ {\displaystyle \alpha } is a root of $m$ {\displaystyle m}. Also, sometimes $m$ {\displaystyle m} can be used to calculate the exponential of $T$ {\displaystyle T} efficiently.

The polynomial calculus is not as informative in the infinite-dimensional case. Consider the unilateral shift with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to spectral theory, since for a diagonal matrix or multiplication operator, it is rather clear what the definitions should be.

References

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"Functional calculus", Encyclopedia of Mathematics , EMS Press, 2001 [1994]

External links

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Media related to Functional calculus at Wikimedia Commons

v t e Analysis in topological vector spaces
Basic concepts	Abstract Wiener space Classical Wiener space Bochner space Convex series Cylinder set measure Infinite-dimensional vector function Matrix calculus Vector calculus
Derivatives	Differentiable vector-valued functions from Euclidean space Differentiation in Fréchet spaces Fréchet derivative Total Functional derivative Gateaux derivative Directional Generalizations of the derivative Hadamard derivative Holomorphic Quasi-derivative
Measurability	Besov measure Cylinder set measure Canonical Gaussian Classical Wiener measure Measure like set functions infinite-dimensional Gaussian measure Projection-valued Vector Bochner / Weakly / Strongly measurable function Radonifying function
Integrals	Bochner Direct integral Dunford Gelfand–Pettis/Weak Regulated Paley–Wiener
Results	Cameron–Martin theorem Inverse function theorem Nash–Moser theorem Feldman–Hájek theorem No infinite-dimensional Lebesgue measure Sazonov's theorem Structure theorem for Gaussian measures
Related	Crinkled arc Covariance operator
Functional calculus	Borel functional calculus Continuous functional calculus Holomorphic functional calculus
Applications	Banach manifold (bundle) Convenient vector space Choquet theory Fréchet manifold Hilbert manifold

v t e Spectral theory and ^*-algebras
Basic concepts	Involution/-algebra Banach algebra B-algebra C-algebra Noncommutative topology Projection-valued measure Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Gelfand representation Polar decomposition Singular value decomposition Spectral theorem Spectral theory of normal C*-algebras
Special Elements/Operators	Isospectral Normal operator Hermitian/Self-adjoint operator Unitary operator Unit
Spectrum	Krein–Rutman theorem Normal eigenvalue Spectrum of a C*-algebra Spectral radius Spectral asymmetry Spectral gap
Decomposition	Decomposition of a spectrum Continuous Point Residual Approximate point Compression Direct integral Discrete Spectral abscissa
Spectral Theorem	Borel functional calculus Min-max theorem Positive operator-valued measure Projection-valued measure Riesz projector Rigged Hilbert space Spectral theorem Spectral theory of compact operators Spectral theory of normal C*-algebras
Special algebras	Amenable Banach algebra With an Approximate identity Banach function algebra Disk algebra Nuclear C*-algebra Uniform algebra Von Neumann algebra Tomita–Takesaki theory
Finite-Dimensional	Alon–Boppana bound Bauer–Fike theorem Numerical range Schur–Horn theorem
Generalizations	Dirac spectrum Essential spectrum Pseudospectrum Structure space (Shilov boundary)
Miscellaneous	Abstract index group Banach algebra cohomology Cohen–Hewitt factorization theorem Extensions of symmetric operators Fredholm theory Limiting absorption principle Schröder–Bernstein theorems for operator algebras Sherman–Takeda theorem Unbounded operator
Examples	Wiener algebra
Applications	Almost Mathieu operator Corona theorem Hearing the shape of a drum (Dirichlet eigenvalue) Heat kernel Kuznetsov trace formula Lax pair Proto-value function Ramanujan graph Rayleigh–Faber–Krahn inequality Spectral geometry Spectral method Spectral theory of ordinary differential equations Sturm–Liouville theory Superstrong approximation Transfer operator Transform theory Weyl law Wiener–Khinchin theorem

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Functional analysis (topics – glossary)

Spaces

Banach Besov Fréchet Hilbert Hölder Nuclear Orlicz Schwartz Sobolev Topological vector
Properties	Barrelled Complete Dual (Algebraic / Topological) Locally convex Reflexive Separable

Theorems

Operators

Algebras

Open problems

Applications

Advanced topics

Category

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See also

References

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