Compact AC(σ) Operators

Abstract

All compact AC(σ) operators have a representation analogous to that for compact normal operators. As a partial converse we obtain conditions which allow one to construct a large number of such operators. Using the results in the paper, we answer a number of questions about the decomposition of a compact AC(σ) operator into real and imaginary parts. (2000). 47B40. Mathematics Subject Classification

Isomorphisms of $$AC(\sigma )$$ spaces for linear graphs

“…This class includes all scalar-type spectral operators, and in. the Hilbert space case, all normal operators (see [AD4]). It is easy to check that if T is an AC(σ) operator then T has spectrum σ(T ) ⊆ σ.…”

Isomorphisms of $BV(σ)$ spaces

“…A bounded operator on a Banach space X which admits an AC(σ) functional calculus is called an AC(σ) operator. The properties of these operators were studied in [5]. At least on reflexive spaces, the AC(σ) spaces play a corresponding role in the spectral theory of AC(σ) operators to that played by C(σ) spaces in the theory of normal operators on Hilbert space.…”

Isomorphisms of AC(σ) spaces

“…Quite naturally then, underlying many of the open problems in this area are questions which ask for analogues of the classical topological results about C(Ω) spaces. (Details of the theory of AC(σ) operators can be found in [5]. ) One of the most classical of these topological results is the Banach-Stone theorem which says that two compact Hausdorff spaces Ω 1 and Ω 2 are homeomorphic if and only if the function algebras C(Ω 1 ) and C(Ω 2 ) are linearly isometric.…”

“…Quite naturally then, underlying many of the open problems in this area are questions which ask for analogues of the classical topological results about C(Ω) spaces. (Details of the theory of AC(σ) operators can be found in [5]. )…”