Functional calculus

Functional calculi are an important mathematical tool for the study of Banach algebras. In the context of operator theory here is especially the Banach algebra of bounded linear operators of interest. For the treatment of unbounded linear operators generalized functional calculi are considered, in which, although basic algebraic structures lost, but which nevertheless provide an effective tool for computing with unbounded operators.

Is a complex polynomial and an element of a Banach algebra with identity - so you can put it in the polynomial by setting. The basic idea of ​​functional calculi is to extend this insertion into polynomials on larger classes of functions. For arbitrary - Banach algebras with unit element, an element in holomorphic functions defined in a neighborhood of the spectrum of can be used. For even larger classes of functions, such as continuous or measurable functions, which are explained on the spectrum of, you have to be limited to special classes of Banach algebras, on C *-algebras and von Neumann algebras. This of course needs to be explained what is meant in this insertion functions at all.

Polynomials

Elements of a Banach algebra with identity - can, as mentioned in the introduction, are inserted directly into polynomials. Are polynomials, the following applies

.

Note the different roles of the plus sign; on the left hand side polynomials are added on the right side elements of a Banach algebra. According applies

,

.

Identifies the range of, the spectral mapping theorem applies

.

On the left side of this formula is the spectrum of the Banach algebras element, on the right side is the image of the spectrum from under the polynomial mapping. The proof of the spectral mapping theorem uses essential that non-constant polynomials have a root, ie it is used the fundamental theorem of algebra. This explains the restriction on - Banach algebras.

This situation is well known in linear algebra. In the investigation of diagonalizability or Jordan normal form also be Banach algebras elements, namely square matrices used in polynomials. For example, states of the Cayley - Hamilton that gives the zero matrix, if one uses a square matrix into its own characteristic polynomial.

For the expansion of onset for major functional classes, we consider the insertion of polynomials in as picture

.

Then is an algebra homomorphism, called the Einsetzungshomomorphismus, and it applies as well. If one has reversed such a homomorphism of a larger class of functions in the Banach algebra A and is a function of this class, so you can define the setting of the function by the formula.

Functional calculi

The further elaboration of the ideas presented here leads to different functional calculi, which are named after the function class used. As a plausible rule of thumb one can say that the situations in which corresponding functional calculi can be used, be specific with increasing functional classes. Typical application examples are discussed in the articles on the individual functional calculi.

  • Holomorphic functional calculus for arbitrary Banach algebras
  • Holomorphic functional calculus of several variables for commutative Banach algebras
  • Continuous functional calculus for C *-algebras
  • Limited Borel functional calculus for von Neumann algebras
  • Unlimited Borel functional calculus for densely - defined self-adjoint operators

Swell

  • J. Dixmier, C * Les algebres et leurs représentations, Gauthier -Villars, 1969
  • R.V. Kadison, JR Ringrose, Fundamentals of the Theory of Operator Algebras, 1983, ISBN 0123933013
  • M. Takesaki, Theory of Operator Algebras I ( Springer 1979, 2002)
  • Functional Analysis
Mathematics C*-Algebra Diagonalmatrix Cayley–Hamilton theorem
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