Multiple sampling and interpolation in Bergman spaces

In this paper we consider linear operators M ψ C ϕ D and M ψ DC ϕ acting between weighted Bergman spaces, where M ψ , C ϕ and D are multiplication, composition and differentation operators respectively.Our goal is to characterize those holomorphic self maps ϕ of D for which M ψ DC ϕ and M ψ C ϕ D acts boundedly and compactly between weighted Bergman spaces.

The commutative C*-algebras of Toeplitz operators on the classical (weightless) Bergman space were classified in Chapter 9 by pencils of geodesics on the unit disk, considered as the hyperbolic plane. Theorem 10.4.1 shows that the same classes of defining symbols generate commutative C*-algebras of Toeplitz operators on each weighted Bergman space. At the same time the principal question, whether the above cases are the only possible sets of defining symbols which might generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space, has remained open.

Norm Approximation by Polynomials in Some Weighted Bergman Spaces

Let L2 = L2(D, r dr dθ/π) be the Lebesgue space on the open unit disc and let be the Bergman space. Let P be the orthogonal projection of L2 onto and let Q be the orthogonal projection onto . Then I − P ≥ Q. The big Hankel operator and the small Hankel operator on are defined as: for ϕ in L∞, and . In this paper, the finite‐rank intermediate Hankel operators between and are studied. We are working on the more general space, that is, the weighted Bergman space.

We show that the Korenblum maximum (domination) principle is valid for weighted Bergman spaces Awp with arbitrary (non-negative and integrable) radial weights w in the case 1≤p<∞. We also notice that in every weighted Bergman space the supremum of all radii for which the principle holds is strictly smaller than one. Under the mild additional assumption lim infr→0+w(r)>00$$\\end{document}]]>, we show that the principle fails whenever 0<p<1.

Let A p α (B n ; C d ) be the weighted Bergman space on the unit ball B n of C n of functions taking values in C d .For 1 < p < ∞ let T p,α be the algebra generated by finite sums of finite products of Toeplitz operators with bounded matrix-valued symbols (this is called the Toeplitz algebra in the case d = 1).We show that every S ∈ T p,α can be approximated by localized operators.This will be used to obtain several equivalent expressions for the essential norm of operators in T p,α .We then use this to characterize compact operators in A p α (B n ; C d ).The main result generalizes previous results and states that an operator in A p α (B n ; C d ) is compact if only if it is in T p,α and its Berezin transform vanishes on the boundary.

The aim of this paper is to show that univalent functions in several classical function spaces can be characterized by integral conditions involving the maximum modulus function. For a suitable choice of parameters, the established condition or its appropriate variant reduces to a known characterization of univalent functions in the Hardy or weighted Bergman space and gives a new characterization of univalent functions in several Mobius invariant function spaces, such as BMOA, Qp or the Bloch space. It is proved, for example, that univalent functions in the Dirichlet type space \( \mathcal{D}_{p + \alpha }^p \) are the same as the univalent functions in Hαp and Sαp if p ≥ 2. Moreover, it is shown that there is in a sense a much smaller Mobius invariant subspace of the Bloch space than Qp still containing all univalent Bloch functions.

Isometric weighted composition operators on weighted Bergman spaces

This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.

We characterize bounded and compact weighted composition operators acting between weighted Bergman spaces and between Hardy spaces. Our results use certain integral transforms that generalize the Berezin transform. We also estimate the essential norms of these operators. As applications, we characterize bounded and compact pointwise multiplication operators between weighted Bergman spaces and estimate their essential norms.

This paper gives the complete proof of the Conjecture given by Hazarika and this author jointly which deals with a necessary and sufficient condition for the hyponormality of Toeplitz operator, T_varphi on the weighted Bergman space with certain polynomial symbols under some assumptions about the Fourier coefficients of the symbol varphi .

A set E c Q is holomorphically dominating for Q if sUP,cE If(z)I = SUpzcn If(z)l for all holomorphic functions f on Q. As follows from a result of Stray, this property is equivalent to the inaccessibility of the Aleksandrov compactification point * (of Q) from Q x E. Moreover, it is equivalent to a large number of other statements (old and new) of holomorphic, harmonic and topological nature, including that a certain weighted Bergman space with p = oo is a Banach space. We extend this to the cases of harmonic functions in R' and holomorphic functions in cV. We also present some results on when weighted Bergman spaces are (quasi)-Banach spaces, the case p = oo being characterised by the result mentioned above.

We study when a weighted composition operator acting between different weighted Bergman spaces is bounded, resp. compact.

We study the boundedness of weighted composition operators acting between weighted Bergman spaces.

We classify self-adjoint first-order differential operators on weighted Bergman spaces on the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are the discrete series representations of $$\textrm{SU}(1,1)$$ . This approach has the promise to generalize to other bounded symmetric domains.