The Interior of Bounded Point Evaluations for Rationally Cyclic Operators

Abstract

Let T be a bounded rationally multicyclic operator on some separable Banach space X, B(T ) be the set of its bounded point evaluations and Ba(T ) be the set of its analytic bounded point evaluations. J. B. Conway asked if the interior of B(T ) and Ba(T ) coincide for arbitrary subnormal operators on Hilbert spaces. Here, we are interested in Conway's problem. We provide an example that answers negatively Conway's question in the more general setting of operators satisfying Bishop's property (β), and we show that Ba(T )\Λ = int(B(T ))\Λ, with different subsets Λ in σ(T ).Mathematics Subject Classification. Primary 47B.

Bounded point evaluation for operators with the wandering subspace property

“…Later, Mbekhta-Ourchane-Zerouali [12] extended and studied the notion of bounded point evaluation for rationally multicyclic operators. A contribution in the study of bounded point evaluations for a cyclic operator was also made by Bourhim-Chidume-Zerouali [3].…”

Spectral picture for rationally multicyclic subnormal operators

“…If E(λ) is bounded from K to (C N , . 1,N ), where x 1,N = N i=1 |xi| for x ∈ C N , then every component in the right hand side extends to a bounded linear functional on H and we will call λ a bounded point evaluation for S. We use bpe(S) to denote the set of bounded point evaluations for S. The set bpe(S) does not depend on the choices of cyclic vectors F1, F2, ..., FN (see Corollary 1.1 in Mbekhta et al (2016)). A point λ0 ∈ int(bpe(S)) is called an analytic bounded point evaluation for S if there is a neighborhood B(λ0, δ) ⊂ bpe(S) of λ0 such that E(λ) is analytic as a function of λ on B(λ0, δ) (equivalently (1-5) is uniformly bounded for λ ∈ B(λ0, δ)).…”

Bounded point evaluations for rationally multicyclic subnormal operators

“…|xi| for x ∈ C N , then every component in the right hand side extends to a bounded linear functional on H and we will call λ a bounded point evaluation for S. We use bpe(S) to denote the set of bounded point evaluations for S. The set bpe(S) does not depend on the choices of cyclic vectors F1, F2, ..., FN (see Corollary 1.1 in Mbekhta et al (2016)). A point λ0 ∈ int(bpe(S)) is called an analytic bounded point evaluation for S if there is a neighborhood B(λ0, δ) ⊂ bpe(S) of λ0 such that E(λ) is analytic as a function of λ on B(λ0, δ) (equivalently (1-7) is uniformly bounded for λ ∈ B(λ0, δ)).…”

“…A point λ0 ∈ int(bpe(S)) is called an analytic bounded point evaluation for S if there is a neighborhood B(λ0, δ) ⊂ bpe(S) of λ0 such that E(λ) is analytic as a function of λ on B(λ0, δ) (equivalently (1-7) is uniformly bounded for λ ∈ B(λ0, δ)). We use abpe(S) to denote the set of analytic bounded point evaluations for S. The set abpe(S) does not depend on the choices of cyclic vectors F1, F2, ..., FN (also see Remark 3.1 in Mbekhta et al (2016)). Similarly, for an N −cyclic subnormal operator S, we can define bpe(S) and abpe(S) if we replace r1, r2, ..., rN ∈ Rat(σ(S)) in (1-7) by p1, p2, ..., pN ∈ P. Bercovici et al (1985) show that the Bergman shift has invariant subspaces with the codimension N property for every N ∈ {1, 2, ..., ∞}.…”

“…There are some results related to Conway's question for rationally multicyclic operators on Banach spaces satisfying Bishops property (β). For example, Mbekhta et al (2016) provides an example of a rationally multicyclic operator T satisfying Bishops property (β), but abpe(T ) = Int(bpe(T )). Also Miller et al (2005) studies rationally cyclic operators.…”