Vol. 24, No. 1, 1968

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Operator valued analytic functions and generalizations of spectral theory

Lothrop Mittenthal

Vol. 24 (1968), No. 1, 119–132

This paper is concerned with an analytic operator valued function F(λ) acting upon a Banach space X, where F(λ) is bounded and F(λ)F(μ) = F(μ)F(λ) for all λ,μ Δ where Δ is the domain of analyticity of F(λ). The singular set of F(λ) is analogous to the spectrum of a single operator. In the case of the single operator, employing the corresponding resolvent operator, a number of interesting properties are known to be associated with the spectral sets. These include projections and homomorphisms between scalar valued analytic functions and functions of the operator. This paper considers a suitable generalization of the resolvent operator and which properties of spectral sets carry over to open and closed subsets of the singular set of the operator valued analytic function.

Mathematical Subject Classification
Primary: 47.30
Received: 22 November 1966
Published: 1 January 1968
Lothrop Mittenthal