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.

Unitary equivalence of operator-valued multishifts

We systematically study various aspects of operator-valued multishifts. Beginning with basic properties, we show that the class of multishifts on the directed Cartesian product of rooted directed trees is contained in that of operator-valued multishifts. Further, we establish circularity, analyticity and wandering subspace property of these multishifts. In the rest part of the paper, we study the function theoretic behaviour of operator-valued multishifts. We determine the bounded point evaluation, reproducing kernel structure and the unitary equivalence of operator-valued multishifts with invertible operator weights. In contrast with a result of Lubin, it appears that the set of all bounded point evaluations of an operator-valued multishift may be properly contained in the joint point spectrum of the adjoint of underlying multishift.

A Characterization of Polynomial Density on Curves via Matrix Algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L 2 ( μ ) , with μ a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure μ . To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, γ ( M ) and λ ( M ) , associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index γ and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index γ that will allow us to give an alternative proof of Thomson’s theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices.

Bounded point evaluations for certain polynomial and rational modules

Let K be a compact subset of the complex plane C . Let P ( K ) and R ( K ) be the closures in C ( K ) of analytic polynomials and rational functions with poles off K , respectively. Let A ( K ) ⊂ C ( K ) be the algebra of functions that are analytic in the interior of K . For 1 ≤ t < ∞ , let P t ( 1 , ϕ 1 , . . . , ϕ N , K ) be the closure of P ( K ) + P ( K ) ϕ 1 + . . . + P ( K ) ϕ N in L t ( d A | K ) , where d A | K is the area measure restricted to K and ϕ 1 , . . . , ϕ N ∈ L t ( d A | K ) . Let H P ( ϕ 1 , . . . , ϕ N , K ) be the closure of P ( K ) ϕ 1 + . . . + P ( K ) ϕ N + R ( K ) in C ( K ) , where ϕ 1 , . . . , ϕ N ∈ C ( K ) . In this paper, we prove if R ( K ) ≠ C ( K ) , then there exists an analytic bounded point evaluation for both P t ( 1 , ϕ 1 , . . . , ϕ N , K ) and H P ( ϕ 1 , . . . , ϕ N , K ) for certain smooth functions ϕ 1 , . . . , ϕ N , in particular, for z ¯ , z ¯ 2 , . . . , z ¯ N . We show that A ( K ) ⊂ H P ( z ¯ , z ¯ 2 , . . . , z ¯ N , K ) if and only if R ( K ) = A ( K ) . In particular, C ( K ) ≠ H P ( z ¯ , z ¯ 2 , . . . , z ¯ N , K ) unless R ( K ) = C ( K ) . We also give an example of K showing the results are not valid if we replace z ¯ n by certain ϕ n , that is, there exist K and a function ϕ ∈ A ( K ) such that R ( K ) ≠ A ( K ) , but A ( K ) = H P ( ϕ , K ) .

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).

A note on 𝐿^{𝑝}-bounded point evaluations for polynomials

We construct a compact nowhere dense subset K K of the closed unit disk D ¯ \bar {\mathbb D} in the complex plane C \mathbb C such that R ( K ) = C ( K ) R(K) = C(K) and bounded point evaluations for P t ( d A | K ) , 1 ≤ t &gt; ∞ , P^t(dA | _K), ~ 1 \le t &gt; \infty , is the open unit disk D . \mathbb D. In fact, there exists C = C ( t ) &gt; 0 C=C(t) &gt; 0 such that \[ ∫ D | p | t d A ≤ C ∫ K | p | t d A , \ \int _{\mathbb D} |p|^t dA \le C \int _K |p|^t dA, \] for 1 ≤ t &gt; ∞ 1 \le t &gt; \infty and all polynomials p . p.

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).

$L^{p}$-bounded point evaluations for polynomials and uniform rational approximation

$L^{p}$-bounded point evaluations for polynomials and uniform rational approximation

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)

Point Evaluations and Polynomial Approximation in the Mean

We construct an example of a Borel subset E of the unit disk \(\{z : \mid z \mid < 1\}\) such that area measure restricted to E, which we denote AE, has the property that the set of bounded point evaluations for the polynomials with respect to the Lt(AE) norm varies with t. We further show that E can be chosen to be a simply connected region. In the context of smooth measures, like area measure, examples of this type were unexpected.

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;

Bounded point evaluations and cyclic polynomials on the spaces $$\hat F$$

LetF be aBK space withAK and $$\hat F$$ denote the set of all formal power series $$\hat f$$ with $$\hat f(z) = \sum\limits_{k = 0}^\infty {f_k z^k } $$ such that $$f = (f_k )_{k = 0}^\infty $$ e F for the sequence of coefficients of $$\hat f$$ . We give a necessary and sufficient condition for a point to be a bounded point evaluation on $$\hat F$$ , and for a polynomial to be cyclic in $$\hat F$$ . As special cases, we obtain the results for the space l p (β) in [7].

The Density Problem for Unbounded Bergman Operators

The unbounded Bergman operator, the operator of multiplication by \( z \) on an unbounded open subset of the plane, is considered. We give a complete answer regarding the density problem of unbounded Bergman operators in terms of its equivalence to the problem of bounded point evaluations for the Bergman spaces. Using this equivalence and the notion of Wiener capacity, we obtain simple geometric conditions that classify almost those open subsets of the plane for which the corresponding Bergman operators are densely defined. With the aid of an analytic approach, we are also able to give condition for a large collection of open subsets of the plane for which all the positive integer powers of the corresponding Bergman operators are densely defined.

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.

Bounded point evaluations for cyclic operators and local spectra

In this paper we study the concept of bounded point evaluations for cyclic operators. We give a negative answer to a question of L.R. Williams, Dynamic Systems and Applications 3 (1994), 103-112. Furthermore, we generalize some results of Williams and give a simple proof that nonnormal hyponormal weighted shifts have fat local 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.