On Nontangential Limits and Shift Invariant Subspaces

Abstract

In 1998, John B. Conway and Liming Yang wrote a paper [11] in which they posed a number of open questions regarding the shift on P t (µ) spaces. A few of these have been completely resolved, while at least one remains wide open. In this paper, we review some of the solutions, mention some alternate approaches and discuss further the problem that remains unsolved.2010 Mathematics Subject Classification. Primary 47A15; Secondary 30C85, 31A15, 46E15, 47B38.

Reproducing kernel of the space $R^t(K,μ)$

“…Theorem 1.3 is a direct application of Theorem 3.6 in [1], which proves a generalized Plemelj's formula for a compactly supported finite complexvalued measure. In fact, the generalized Plemelj's formula holds for rectifiable curve (other than T), so Theorem 1.3 is valid if ∂Ω is a certain rectifiable curve.…”

“…The proof is excellent but complicated, and it does not really lend itself to showing the existence of nontangential boundary values in the case that spt(µ) ⊆ D, P t (µ) is irreducible and µ(T) > 0. X. Tolsa's remarkable results on analytic capacity opened the door for a new view of things, through the works of [1], [2], [3] and [4], etc.…”

“…It is known that ∂ so,σ Ω is a Borel set (i.e., see Lemma 4 in [9]) and m(∂ so,σ1 Ω\ ∂ so,σ2 Ω) = 0 for σ 1 = σ 2 . Therefore, we set ∂ so Ω = ∂ so, 1 2 Ω. The paper [1] presents an alternate and simpler route to prove Theorem 1.2 (a) and (b) that has extension to the context of mean rational approximation as in Theorem 1.3 below.…”

“…Therefore, we set ∂ so Ω = ∂ so, 1 2 Ω. The paper [1] presents an alternate and simpler route to prove Theorem 1.2 (a) and (b) that has extension to the context of mean rational approximation as in Theorem 1.3 below. It also uses the results of X. Tolsa on analytic capacity.…”

Approximation in the mean by rational functions II

“…For a compactly supported complex Borel measure ν of C, by estimating analytic capacity of the set {λ : |C(ν)| > c}, J. Brennan [B06], A. Aleman, S. Richter, and C. Sundberg [ARS09] and [ARS10] provide interesting alternative proofs of Thomson's theorem for the existence of analytic bounded point evaluations for mean polynomial approximation. Using X. Tolsa's deep results on analytic capacity and Cauchy transform, J. Akeroyd, J. Conway, L.Yang [ACY19] develops a Plemelj's formula for an arbitrary measure and generalizes Theorem A of [ARS09] to certain R t (K, µ). J. Conway and L. Yang [CY19] introduce the key concept of non-removable boundary and obtains structural results for R t (K, µ).…”