Analytic Bounded Point Evaluations Research Articles

Approximation in the Mean on Rational Curves

In the presence of a positive, compactly supported measure on an affine algebraic curve, we relate the density of polynomials in Lebesgue \(L^2\)-space to the existence of analytic bounded point evaluations. Analogues to the complex plane results of Thomson and Brennan are obtained on rational curves.

Bounded point evaluation for a finitely multicyclic commuting tuple of operators

We generalize the notion of bounded point evaluation introduced by Williams for a cyclic operator to a finitely multicyclic commuting d-tuple T=(T1,…,Td) of bounded linear operators on a complex separable Hilbert space. We show that the set bpe(T) of all bounded point evaluations for T is a unitary invariant and we characterize it in terms of the dimension of the joint cokernel of T. Using this, we show that if bpe(T) has non-empty interior, then T can be realized as the d-tuple Mz=(Mz1,…,Mzd) of multiplication operators on a reproducing kernel Hilbert space H of functions on bpe(T). We further characterize the largest open subset of bpe(T) on which all the elements of H are analytic, which we refer to as the set of all analytic bounded point evaluations. As an application, we describe the set of all analytic bounded point evaluations for toral and spherical isometries, and also, derive an analytic model of a commuting d-tuple of composition operators.

Spectral picture for rationally multicyclic subnormal operators

For a pure bounded rationally cyclic subnormal operator $S$ on a separable complex Hilbert space $\mathcal H,$ J. B. Conway and N. Elias (Analytic bounded point evaluations for spaces of rational functions, J. Functional Analysis, 117:1{24, 1993) showed that $clos(\sigma (S) \setminus \sigma_e (S)) = clos(Int (\sigma (S))).$ This paper examines the property for rationally multicyclic (N-cyclic) subnormal operators. We show: (1) There exists a 2-cyclic irreducible subnormal operator $S$ with $clos(\sigma (S) \setminus \sigma_e (S)) \neq clos(Int (\sigma (S))).$ (2) For a pure rationally $N-$cyclic subnormal operator $S$ on $\mathcal H$ with the minimal normal extension $M$ on $\mathcal K \supset \mathcal H,$ let $\mathcal K_m = clos (span\{(M^*)^kx: ~x\in\mathcal H,~0\le k \le m\}.$ Suppose $M |_{\mathcal K_{N-1}}$ is pure, then $clos(\sigma (S) \setminus \sigma_e (S)) = clos(Int (\sigma (S))).$

Bounded point evaluations for rationally multicyclic subnormal operators

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)∖σe(S).

The Interior of Bounded Point Evaluations for Rationally Cyclic Operators

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 B a (T) be the set of its analytic bounded point evaluations. J. B. Conway asked if the interior of B(T) and B a (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 $${(\beta)}$$ , and we show that $${B_a(T) {\setminus} \Lambda = {\rm int}(B(T)){\setminus} \Lambda }$$ , with different subsets Λ in $${\sigma}$$ (T).

Nontangential limits in 𝒫t(μ)-spaces and the index of invariant subgroups

Let mu be a finite positive measure on the closed disk (D) over bar in the complex plane, let 1 <= t < infinity, and let P-t(mu) denote the closure of the analytic polynomials in L-t(mu). We suppose that D is the set of analytic bounded point evaluations for P-t(mu), and that P-t(mu) contains no nontrivial characteristic functions. It is then known that the restriction of mu to partial derivative D must be of the form h vertical bar dz vertical bar. We prove that every function f is an element of P-t(mu) has nontangential limits at h vertical bar dz vertical bar-almost every point of partial derivative D, and the resulting boundary function agrees with f as an element of L-t(h vertical bar dz vertical bar). Our proof combines methods from James E. Thomson's proof of the existence of bounded point evaluations for P-t(mu) whenever P-t(mu) not equal L-t(mu) with Xavier Tolsa's remarkable recent results on analytic capacity. These methods allow us to refine Thomson's results somewhat. In fact, for a general compactly supported measure nu in the complex plane we are able to describe locations of bounded point evaluations for P-t(nu) in terms of the Cauchy transform of an annihilating measure. As a consequence of our result we answer in the affirmative a conjecture of Conway and Yang. We show that for 1 < t < infinity dim M/zM = 1 for every nonzero invariant subspace M of P-t(mu) if and only if h not equal 0. We also investigate the boundary behaviour of the functions in P-t(mu) near the points z is an element of partial derivative D where h(z) = 0. In particular, for 1 < t < infinity we show that there are interpolating sequences for P-t(mu) that accumulate nontangentially almost everywhere on {z : h(z) = 0}. (Less)

Analytic Bounded Point Evaluations for Rationally Cyclic Operators on Banach Spaces

In the general context of rationally cyclic operators on Banach spaces, this article centers around descriptions of the set of analytic bounded point evaluations in the spirit of function theoretic operator theory. In particular, a classical formula due to Trent for subnormal cyclic operators is extended to the case of subdecomposable operators, based on tools from local spectral theory and the Kato spectrum. Particular emphasis is placed on the connection to localized versions of the single-valued extension property and Bishop’s property (β). The results are exemplified in the case of hyponormal operators, unilateral and bilateral weighted shifts, and the Cesaro operator on the Bergman space.

Bounded point evaluations for cyclic Hilbert space operators

&lt;p&gt;In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.&lt;/p&gt;

On the largest analytic set for cyclic operators

We describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space ℋ; some related consequences are discussed. Furthermore, we show that two densely similar cyclic Banach‐space operators possessing Bishop′s property (β) have equal approximate point spectra.

The commutant of rationally cyclic subnormal operators and rational approximation

Let Ω denote the set of analytic bounded point evaluations forR q (K, μ). Assume that $$\bar \Omega = K$$ . In this paper, we first show that if Ω is a finitely connected domain and if the evaluation map fromR q (K, μ)пL ∞(μ) toH ∞(Ω) is surjective, then μ|∂Ω is absolutely continuous with respect to harmonic measure for Ω. This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when Ω is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.

The Berger-Shaw theorem for cyclic subnormal operators

This work gives a sharp form of the Berger-Shaw Theorem for cyclic subnormal operators. That is, if S is a cyclic subnormal operator, then S has trace class self- commutator and the trace equals (1=…)Area(ae(S)iaee(S)). It also characterizes those functions f such that f(S) has trace class self-commutator and computes the trace as the Dirichlet integral of ^ f on the set of analytic bounded point evaluations for S. The technique used also gives some partial results for rationally cyclic subnormal operators.

Approximation in the mean by rational functions

In this paper we proved that for a large class of compact subsetsK in the complex plane,R(K) is dense inLq(μ) if and only if the set of analytic bounded point evaluations forRq(K, μ) is empty. As a consequence, we showed that this result is true for allK ifR(K) is replaced byA(K). Our main result includes the corresponding result of James Thomson for polynomials approximation as such a special case thatK is a disk.

Analytic Bounded Point Evaluations for Spaces of Rational Functions

For a compact subset of the complex plane K and a regular Borel measure μ supported on K, R p ( K, μ) denotes the closure in L p (μ) of thè rational functions with poles off K. This paper examines the existence of analytic bounded point evaluations on R p ( K, μ) and extends the work of Thomson on the existence of analytic bounded point evaluations for the closure in L p (μ) of the analytic polynomials. Provided R p ( K, μ) is "pure," it is shown that the closure of the set of analytic bounded point evaluations for R p ( K, μ) equals the closure of the interior of the spectrum of the operator multiplication by z on R p ( K, μ). In fact this is derived as a consequence of Thomson′s result for polynomials. The bulk of the paper is devoted to certain interpolation problems for R p ( K, μ), thus producing some information on the structure of R p ( K, μ).

Sufficient conditions for bounded point evaluations

A geometric condition for the existence of bounded evaluations is given. Using this criterion for sets of finite perimeter, it is shown that P2(wdm2) has analytic bounded point evaluations, when w is of bounded variation with compact support. This theorem is then related to previous work on bounded point evaluations.

Analytic structures for subnormal operators

For μ a compactly supported measure on ℂ, we construct a mutually absolutely continuous measure ν so thatP2(ν) has analytic bounded point evaluations, and the operator of multiplication byz onP2(ν) has every invariant subspace hyperinvariant. We also construct an equivalent measure σ so thatR2(K, σ) has as analytic bounded point evaluations precisely the interior of the set of weak-star continuous point evaluations ofR∞(K, μ). In the course of the proof, we classify weak-star closed super-algebras ofR∞(K, μ) whenR(K) is hypo-Dirichlet.

A structure theorem for the commutant of a class of cyclic subnormal operators

An m m -measure is defined to be a measure μ \mu such that the analytic bounded point evaluations of P 2 ( μ ) {P^2}(\mu ) is the open unit disk D {\mathbf {D}} in the complex plane, and the weak* closure of the analytic polynomials in L ∞ ( μ ) {L^\infty }(\mu ) is the set of bounded analytic functions on D {\mathbf {D}} . A complete characterization of P 2 ( μ ) ∩ L ∞ ( μ ) {P^2}(\mu ) \cap {L^\infty }(\mu ) , the commutant of the cyclic subnormal operator of multiplication by z z on P 2 ( μ ) {P^2}(\mu ) , is then obtained.