Holomorphic functional calculus

The holomorphic functional calculus is a basic method of the mathematical theory of Banach algebras. Roughly speaking, this will be in the functional elements of a calculus in Banach - holomorphic functions that are defined in an area of the spectrum of the element, are used, so that the insertion into the polynomials will be generalized.

Construction

It is a Banach algebra with identity -. If so, the spectrum is non- empty (see Gelfand - Mazur ). Let further a function defined in an open neighborhood of holomorphic function. Although it can not be used directly in, but the Cauchy integral formula gives a representation of the values ​​of the function at which such transfer can be carried out nonetheless.

There is a simple closed cycle paths that include very run and the spectrum. The Cauchy integral formula for points within, and the fact you can actually use the Banach algebras element. It can be shown that the integral

Converges in the sense of the norm topology. Here the term is defined in the integrand and is a continuous function. Next, one can show that this value does not depend on the particular choice of. Therefore, one designated by the value of this integral in suggestive notation.

For a compact set is the set defined in a neighborhood of holomorphic functions. Are and two such functions, so you can and on the average of the domains of and explain. This becomes an algebra. With the above definitions we obtain a picture. This mapping is called the holomorphic functional calculus of a

The requirement that a unit element has, is not a significant limitation, because you can, if necessary, adjoin a unit element and apply the functional calculus in the enlarged Banach algebra.

Properties

The holomorphic functional calculus for an element has the following properties.

• Is a homomorphism, that is, apply the formulas.
• If in a neighborhood of the spectrum is a power series representation, it is deemed absolutely convergent series in.
• If and so true.
• It is the spectral mapping theorem: for all.

One can imagine, actually use the elements in Banach algebras holomorphic functions, ie; the obvious algebraic operations behave as expected.

Application

As a typical application of the holomorphic functional calculus, we prove the following theorem:

For one - Banach algebra with unit element are equivalent:

• Has with projections.
• Has elements with incoherent spectrum.

This statement can be tightened to Schilowschen Idempotentensatz, which requires the underlying holomorphic functional calculus of several variables.