Analytical Functional Models and Local Spectral Theory

Abstract

In 1959 E. Bishop used a Banach‐space version of the analytic duality principle established by e Silva, Köthe, Grothendieck and others to study connections between spectral decomposition properties of a Banach‐space operator and its adjoint. According to Bishop a continuous linear operator T ∈ L(X) on a Banach space X satisfies property (rβ) if the multiplication operator Ofalse(U,Xfalse)false→Ofalse(U,Xfalse),ffalse↦false(z−Tfalse)f, is injective with closed range for each open set U in the complex plane. In the present article the analytic duality principle in its original locally convex form is used to develop a complete duality theory for property (rβ). At the same time it is shown that, up to similarity, property (rβ) characterizes those operators occurring as restrictions of operators decomposable in the sense of C. Foias, and that its dual property, formulated as a spectral decomposition property for the spectral subspaces of the given operator, characterizes those operators occurring as quotients of decomposable operators. It is proved that, unlike the situation for commuting subnormal operators, each finite commuting system of operators with property (rβ) can be extended to a finite commuting system of decomposable operators. Meanwhile the results of this paper have been used to prove the existence of invariant subspaces for subdecomposable operators with sufficiently rich spectrum. 1991 Mathematics Subject Classification: 47A11, 47B40.

Back to RS-SR spectral theory

“…In particular, compact and algebraic operators are decomposable. It is also known that (β) characterizes operators with decomposable extensions and in particular operators with (β) are provided by isometries and subnormal operators [2]. The property (β) is hence conserved by restrictions while (δ) is transferred to quotient operators.…”

“…The property (β) is hence conserved by restrictions while (δ) is transferred to quotient operators. We refer to [1,2,22] for a complete study and further properties and results.…”

The Closed Range Property for Banach Space Operators

“…Moreover, [2,Theorems 8 and 21], T is decomposable on U if and only if T and its adjoint T * share property (β) on U. Thus Nagy's largest open set on which T is decomposable is the set ρ β (T) ∩ ρ β (T * ).…”

“…Fundamental work by Albrecht and Eschmeier established that an operator T ∈ B(X) has property (β) on U precisely when there exists an operator S ∈ B(Y ) such that S is decomposable on U, X ∈ Lat (S) and T = S| X , [2,Theorem 10]. Moreover, [2,Theorems 8 and 21], T is decomposable on U if and only if T and its adjoint T * share property (β) on U.…”

The Cesàro and related operators, a survey

“…Thus, T has property (β) if, whenever (f n ) n is a sequence of analytic X-valued functions such that (λ − T )f n (λ) → 0 uniformly on the compact subsets of an open U ⊆ C, then f n (λ) → 0 uniformly on the compact subsets of U . This seemingly technical property in fact completely characterizes the restrictions of bounded decomposable operators to invariant subspaces, and T ∈ L(X) is decomposable if and only if both T and T * have property (β), [1], [25].…”