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.
