Functional calculus

Functional calculi are an important mathematical tool for studying Banach algebras . In the context of operator theory, the Banach algebra of bounded linear operators is of particular interest. In order to deal with unbounded linear operators , generalized functional calculi are considered, in which basic algebraic structures are lost, but which nevertheless provide an effective tool for computing with unbounded operators.

If a complex polynomial and an element of a -Banach algebra with one element , one can insert into the polynomial by inserting . The basic idea of ​​functional calculi is to extend this insertion into polynomials to larger classes of functions. For any -Banach algebras with one element, an element can be used in holomorphic functions that are defined in a neighborhood of the spectrum of . For even larger functional classes , such as continuous or measurable functions , which are explained on the spectrum of , one has to restrict oneself to special classes of Banach algebras, namely to C ^* algebras or Von Neumann algebras . For this, of course, it must be explained what this insertion in functions is supposed to mean. ${\displaystyle p(z)=\lambda _{n}z^{n}+\ldots +\lambda _{1}z+\lambda _{0}}$ ${\displaystyle a}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle e}$ ${\displaystyle a}$ ${\displaystyle p}$ ${\displaystyle p(a)=\lambda _{n}a^{n}+\ldots +\lambda _{1}a+\lambda _{0}e}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle a\in A}$ ${\displaystyle a}$ ${\displaystyle a}$

Polynomials

As mentioned in the introduction, elements of a -Banach algebra with one element can be inserted directly into polynomials . If there are polynomials, then ${\displaystyle a}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle e}$ ${\displaystyle p\in {\mathbb {C} }[z]}$ ${\displaystyle p,q\in {\mathbb {C} }[z]}$

${\displaystyle (p+q)(a)\,=\,p(a)+q(a)}$.

Note the different roles of the plus sign; on the left side polynomials are added, on the right side elements of a Banach algebra. The same applies accordingly

${\displaystyle (p\cdot q)(a)=p(a)\cdot q(a)}$,

${\displaystyle (p\circ q)(a)=p(q(a))}$.

If the spectrum denotes , then the spectral mapping theorem applies ${\displaystyle \sigma (a)}$ ${\displaystyle a}$

${\displaystyle \sigma (p(a))\,=\,p(\sigma (a))}$.

On the left side of this formula is the spectrum of the Banach algebra element , on the right side is the image of the spectrum from below the polynomial mapping . The proof of the spectral mapping theorem makes substantial use of the fact that non-constant polynomials have a zero; H. the fundamental theorem of algebra is used. This explains the limitation to -Banach algebras. ${\displaystyle p(a)}$ ${\displaystyle a}$ ${\displaystyle p:\mathbb {C} \rightarrow \mathbb {C} }$ ${\displaystyle \mathbb {C} }$

This situation is well known from linear algebra . In the investigation of the diagonalisability or the Jordan normal form , Banach algebra elements, namely square matrices , are also used in polynomials. For example, Cayley-Hamilton's theorem says that the zero matrix is obtained by inserting a square matrix into its own characteristic polynomial .

To extend the insertion to larger functional classes, we consider the insertion of in polynomials as a mapping ${\displaystyle a\in A}$

${\displaystyle \Phi _{a}:{\mathbb {C} }[z]\rightarrow A,\,\,p\mapsto p(a)}$.

Then there is an algebra homomorphism , the so-called insertion homomorphism of , and it holds as well . Conversely, if one has such a homomorphism from a larger functional class into the Banach algebra and is a function of this class, then the insertion of into the function can be defined by the formula . ${\displaystyle \Phi _{a}}$ ${\displaystyle a}$ ${\displaystyle \Phi _{a}(1)=e}$ ${\displaystyle \Phi _{a}(z)=a}$ ${\displaystyle \Phi _{a}}$ ${\displaystyle A}$ ${\displaystyle f}$ ${\displaystyle a}$ ${\displaystyle f}$ ${\displaystyle f(a):=\Phi _{a}(f)}$

Functional calculus

The further elaboration of the ideas presented here leads to different functional calculi, which are named after the functional class used. As a plausible rule of thumb, one can say that the larger the function classes, the more special the situations in which the associated functional calculi can be used. Typical application examples are dealt with 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
• bounded Borel functional calculus for Von Neumann algebras
• unbounded Borel functional calculus for densely-defined self-adjoint operators

Web links

• Functional calculus (Encyclopedia of Mathematics)

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• J. Dixmier , Les C * -algèbres et leurs représentations, Gauthier-Villars, 1969
• RV Kadison, JR Ringrose, Fundamentals of the Theory of Operator Algebras, 1983, ISBN 0123933013
• M. Takesaki, Theory of Operator Algebras I (Springer 1979, 2002)