Abstract
We will consider multiplication operators on a Hilbert space of analytic functions on a domainΩ⊂C. For a bounded analytic functionφonΩ, we will give necessary and sufficient conditions under which the complement of the essential spectrum ofMφinφΩbecomes nonempty and this gives conditions for the adjoint of the multiplication operatorMφbelongs to the Cowen-Douglas class of operators. Also, we characterize the structure of the essential spectrum of a multiplication operator and we determine the commutants of certain multiplication operators. Finally, we investigate the reflexivity of a Cowen-Douglas class operator.
Highlights
We include some preparatory material which will be needed later.For a positive integer n and a domain U ⊂ C, the CowenDouglas class Bn(U) consists of bounded linear operators T on any fixed separable infinite dimensional Hilbert space X with the following properties:(a) U is a subset of the spectrum of T.(b) Ran(λ − T) = X for every λ ∈ U.(c) dim[ker(λ − T)] = n for every λ ∈ U.(d) Span{ker(λ − T) : λ ∈ U} = X.Here Span denotes the closed linear span of a collection of sets in X
By a method used in the proof of [3, Proposition 3.1] we show that the function ψ is bounded below on Ω
We investigate the converse of Theorem 1
Summary
We include some preparatory material which will be needed later. For a positive integer n and a domain U ⊂ C, the CowenDouglas class Bn(U) consists of bounded linear operators T on any fixed separable infinite dimensional Hilbert space X with the following properties:. Recall that a bounded linear operator A on a Hilbert space is a Fredholm operator if and only if ran A is closed and both ker A and ker A∗ are finite dimensional. Let H be a separable Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. If H is a Hilbert space of functions analytic on a plane domain Ω, a complex-valued function φ on Ω for which φf ∈ H for every f ∈ H is called a multiplier of H and the multiplier φ on H determines a multiplication operator Mφ on H by Mφf = φf, f ∈ H. In the rest of the paper we assume that H is a Hilbert space of analytic function on a bounded plane domain Ω. For some other source on these topics one can see [10,11,12,13,14,15,16]
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