The Essential Spectrum and Banach Reducibility of Operator Weighted Shifts

A characterization of bilateral operator weighted shifts being Cowen-Douglas operators

“…It implies that σ(S + ) ⊂ Ω 0 . By Proposition 2.3 in [5], moreover, we have that S − ∈ B n (Ω 0 ). Set Ω = Ω 0 \σ(S + ).…”

“…Hence, S + cannot be a Cowen-Douglas operator. When S * + is a Cowen-Douglas operator has been characterized in [5]. In this note, we will prove the following theorem.…”

“…In [5], σ e (S + ) and the indices associated with all holes in σ e (S + ) have been described. For convenience, we draw the following conclusions.…”

“…By the Riesz decomposition theorem, if σ(T ) is not connected, then T is not strongly irreducible. For a unilateral operator weighted shift S + , it follows from Theorem 3.1 in [5] that if σ e (S + ) is not connected, then S + is not strongly irreducible. For a bilateral operator weighted shift S, however, the following example shows that a similar claim is false even though S is a Cowen-Douglas operator.…”

The single-valued extension property for bilateral operator weighted shifts

“…In [4], Ben-Artzi and Gohberg introduced the concepts of Bohl exponent and canonical splitting projection to describe the spectrum and the essential spectrum of operator weighted shifts of finite multiplicity (see also [10]). However, in the general setting of bilateral operator weighted shifts of infinite multiplicity, the complete description of the spectrum and its parts is not yet settled.…”