An operator satisfying Dunford's condition (C) but without bishop's property (β)

Abstract

1. Introduction. For X a complex Banach space and U an open subset of the complex plane C, let O{U, X) denote the space of analytic X-valued functions defined on U. This is a Frechet space when endowed with the topology of uniform convergence on compact subsets, and the space X may be viewed as simply the constants in O(U, X). Every bounded operator T on X induces a continuous mapping , X) and X e U. Corresponding to each closed F c C there is also an associated analytic subspace X T (F) = XD ran(7c//r). For an arbitrary Te C(X), the spaces XT(F) are T-invariant, generally non-closed linear manifolds in X.An operator T e C(X) has the decomposition property (8) provided that the space X decomposes asis an open cover of the complex plane. T e C(X) is decomposable in the sense of Foias provided that T has property (<5) and that the analytic subspaces XT(F) are closed whenever F is a closed subset of the plane; see [1], [3].The local resolvent of Tata vector x e X is the set PT(X) consisting of all X e C for which there is a open neighborhood U such that x e TyO{U, X). The local spectrum of T at x is a-rix) = C\PT(X).If T is such that for every closed F c C the linear manifold An operator T has Bishop's property (ft) provided that for every open U C C the mapping Tu is injective and has closed range. Albrecht and Eschmeier [2] showed that property (fi) completely characterizes the restrictions of decomposable operators to invariant subspaces and their analytic functional model shows that every Banach space operator is similar to the quotient of an operator with property (/3); see [5]. Moreover, Albrecht and Eschmeier prove properties (P) and (S) to be completely dual; an operator T has one of these precisely when T* has the other.Property {p) implies ( Q , and ( Q in turn implies the single-valued extension property, [6, Proposition 1.2]. Therefore T e C(X) has property (C) if and only if the analytic subspaces XT(F) are closed whenever F is a closed subset of the plane, and T is decomposable if and only if T has both properties (C) and (8); equivalently, if and only if T has both properties (0) and (8). Thus it is a natural question whether property ( Q is strictly weaker than property (()). Laursen and Neumann mention it explicitly in [6], but this question has circulated informally for some time.

On the local spectral properties of weighted shift operators

“…In fact, this result could be obtained by combining Theorem 3.1 of [24] and Theorem 10(ii) of [20]. But the proof given in [16] is simple and elegant. Here, we refine this result as follows.…”

“…In [16], T. L. Miller and V. G. Miller have shown that if r 1 (S) < r 2 (S), then S fails Bishop's property (β). In fact, this result could be obtained by combining Theorem 3.1 of [24] and Theorem 10(ii) of [20].…”

Sur la théorie spectrale locale des shifts à poidsopérateurs

“…L'étude de la théorie spectrale locale des opérateurs shiftsà poids scalaires a fait l'objet de plusieurs travaux, on cite par exemple les références [4,13,14,16,17]. Cependant, il apparaît qu'aucuneétude satisfaisante des propriétés spectrales locales d'un shiftà poids opérateurs n'est encore faite.…”

“…Dans la dernière section nous complétons et nous généralisons les résultats figurant dans [13]. Nousétablisserons des conditions nécessaires pour qu'un opérateur shift satisfasse la condition de Bishop (β).…”

“…Dans le cas où les poids sont inversibles, le théorème suivant, qui généralise le résultat de [13], donne une condition suffisante qui assure la propriété de Dunford (C). Le résultat est obtenu aussi dans [12, proposition 1.6.5] lorsque r 1 (T) = r(T).…”