Bounded point evaluations for rationally multicyclic subnormal operators

Abstract

Let S be a pure bounded rationally multicyclic subnormal operator on a separable complex Hilbert space H and let Mz be the minimal normal extension on a separable complex Hilbert space K containing H. Let bpe(S) be the set of bounded point evaluations and let abpe(S) be the set of analytic bounded point evaluations. We show abpe(S) = bpe(S) ∩ Int(σ(S)). The result affirmatively answers a question asked by J. B. Conway concerning the equality of the interior of bpe(S) and abpe(S) for a rationally multicyclic subnormal operator S. As a result, if λ0 ∈ Int(σ(S)) and dim(ker(S − λ0) * ) = N, where N is the minimal number of cyclic vectors for S, then the range of S − λ0 is closed, hence, λ0 ∈ σ(S)

On Nontangential Limits and Shift Invariant Subspaces

“…we find that (3.6) and (3.7) hold. On B(λ 0 , δ) \ E δ and for ǫ < δ, we conclude that (1) The above lemma, which is needed in several places of this paper, is a generalization of Lemma 4 in [20].…”

“…The above lemma, which is needed in several places of this paper, is a generalization of Lemma 4 in [20].…”

Spectral picture for rationally multicyclic subnormal operators

“…Corollary 5.2 in Conway and Elias (1993) proves that the result holds for rationally cyclic subnormal operators. For N > 1, Yang (2018) extends the result to rationally N −cyclic subnormal operators.…”

“…Proof: Using (3-4), (3-5), and (3-6), we see that the lemma is a direct application of Theorem 2 in Yang (2018).…”

Bounded point evaluations for certain polynomial and rational modules

“…Both their proofs rely on X. Tolsa's deep results on analytic capacity. Yang (2018) extends some results to a rationally multicyclic subnormal operator (restriction of a normal operator on a separable Hilbert space to an invariant subspace).…”