On Dirichlet Spaces With a Class of Superharmonic Weights

In 1993, N. Danikas and A. G. Siskakis showed that the Cesàro operator ${\mathcal{C}}$ is not bounded on $H^{\infty }$; that is, ${\mathcal{C}}(H^{\infty })\nsubseteq H^{\infty }$, but ${\mathcal{C}}(H^{\infty })$ is a subset of $BMOA$. In 1997, M. Essén and J. Xiao gave that ${\mathcal{C}}(H^{\infty })\subsetneq {\mathcal{Q}}_{p}$ for every $0<p<1$. In this paper, we characterize positive Borel measures $\unicode[STIX]{x1D707}$ such that ${\mathcal{C}}(H^{\infty })\subseteq M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and show that ${\mathcal{C}}(H^{\infty })\subsetneq M({\mathcal{D}}_{\unicode[STIX]{x1D707}_{0}})\subsetneq \bigcap _{0<p<\infty }{\mathcal{Q}}_{p}$ by constructing some measures $\unicode[STIX]{x1D707}_{0}$. Here, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ denotes the Möbius invariant function space generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, where ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ is a Dirichlet space with superharmonic weight induced by a positive Borel measure $\unicode[STIX]{x1D707}$ on the open unit disk. Our conclusions improve results mentioned above.

<abstract><p>In this paper, we investigated the Hyers–Ulam stability of the coefficient multipliers on the Hardy space $ H^2 $ and the Dirichlet space $ \mathcal{D}^2 $. We also investigated the Hyers–Ulam stability of the coefficient multipliers between Dirichlet and Hardy spaces. We provided the necessary and sufficient conditions for the coefficient multipliers to have Hyers–Ulam stability on Hardy space $ H^2 $, on Dirichlet space $ \mathcal{D}^2 $, and between Dirichlet and Hardy spaces. We also showed that the best constant of Hyers–Ulam stability exists under different circumstances. Moreover, some illustrative examples were discussed.</p></abstract>

Let E \subset(-1,1) be a compact set, let μ be a positive Borel measure with support \supp μ =E , and let H p (G),1≤ p ≤∈fty, be the Hardy space of analytic functions on the open unit disk G with circumference Γ={z \colon |z|=1} . Let Δ n,p be the error in best approximation of the Markov function \frac{1}{2π i} ∈t_E \frac{d μ(x)}{z-x} in the space L p (Γ) by meromorphic functions that can be represented in the form h=P/Q , where P ∈ H p (G),Q is a polynomial of degree at most n , Q\not \equiv 0 . We investigate the rate of decrease of Δ n,p ,1≤ p ≤∈fty , and its connection with n -widths. The convergence of the best meromorphic approximants and the limiting distribution of poles of the best approximants are described in the case when 1 - ∈fty.

It is shown that there exist analytic self-maps ϕ of the unit disc inducing compact composition operators on the Hardy space , 1 ≤ p < ∞ such that the Hausdorff dimension of the set is one; sharpening a classical result due to Schwartz. Moreover, the same holds in the weighted Dirichlet spaces with 0 < α < 1. As a consequence, we deduce that there exist symbols ϕ inducing compact composition operators on such that the α-capacity of E ϕ is positive, which is no longer true for those just inducing Hilbert-Schmidt composition operators on .

We generalise previous results of the author concerning the compactness of composition operators on the Hardy spaces $H^{p}$, $1\leq p<\infty$, whose symbol is a universal covering map from the unit disk in the complex plane to general finitely connected domains. We demonstrate that the angular derivative criterion for univalent symbols extends to this more general case. We further show that compactness in this setting is equivalent to compactness of the composition operator induced by a univalent mapping onto the interior of the outer boundary component of the multiply connected domain.

If α>-1-1$$\\end{document}]]> the space of Dirichlet type Dα2 consists of those functions f which are analytic in the unit disc D such that f′ belongs to the weighted Bergman space Aα2. The space D02 is the classical Dirichlet space D. If g is an analytic function in D, we study the generalized Hilbert operator Hg defined by Hg(f)(z)=∫01f(t)g′(tz)dtacting on the spaces Dα2 (0≤α≤1). We obtain a characterization of those g for which Hg is bounded, compact, or Hilbert-Schmidt on the Dirichlet space D. In addition to this, we use our results concerning the operators Hg to study certain Cesàro-type operators C(η) acting on the spaces Dα2 (0≤α≤1). We give also a characterization of the positive finite Borel measures μ in [0, 1) for which a certain Cesàro type operator Cμ associated to μ is bounded on the Bergman space Aα1 (α>-1-1$$\\end{document}]]>). This is an extension of the previously known results for the spaces Aαp with p>11$$\\end{document}]]> and α>-1-1$$\\end{document}]]>.

Composition Operators from Bloch Type Spaces to Hardy and Besov Spaces

We show that a restriction on the angular derivative of the inducing map does not determine compact composition operators on large weighted Hardy spaces, thus answering in the negative a question posed by T. Kriete.

Let X be a set of analytic functions on the open unit disk D, and let phi be an analytic function on D such that phi(D) is contained in D and f |-> f o phi takes X into itself. We present conditions on X ensuring that if f |-> f o phi is invertible on X, then phi is an automorphism of D, and we derive a similar result for mappings of the form f |-> psi.(f o phi), where psi is some analytic function on D. We obtain as corollaries of this purely function-theoretic work, new results concerning invertibility of composition operators and weighted composition operators on Banach spaces of analytic functions such as S^p and the weighted Hardy spaces H^2(beta).

Compact composition operators on generalized Lipschitz spaces

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known ideas. Rather, we completely reorganize the subject in order to bring out the profound contributions of E. M. Stein to the study of the boundary behavior both of holomorphic and harmonic functions in one and several variables. In an appendix, we provide a self-contained proof of a new result which is relevant to the differentiation of integrals, a topic which, as witnessed in Stein’s work, and especially by the aforementioned book, has deep connections with the boundary behavior of harmonic and holomorphic functions.

Abstract. AsequenceofcompositionoperatorsonHardyspaceiscon-sidered. Weprovethat,bynumericalrangeproperties,itisSOT-conver-gencebutnotconverge. 1. introductionLet ϕ be a holomorphic self-map of the unit disc U := {z ∈ C : |z| < 1}.The function ϕ induces the composition operator C ϕ , defined on the space ofholomorphic functions on Uby C ϕ f = f ◦ ϕ. The restriction of C ϕ to variousBanach spaces of holomorphic functions on U has been an active subject ofresearch for more than three decades and it will continue to be for decadesto come (see [11], [12] and [4]). Let H 2 denote the Hardy space of analyticfunctions on the open unit disc with square summable Taylor coefficients. Inrecent yearsthe study ofcomposition operatorson the Hardy space has receivedconsiderable attention.In this paper we consider the numerical range of elliptic composition opera-tors on H 2 . The numerical range of a bounded linear operator A on a Hilbertspace H is the set of complex numbersW(A) := {hAx,xi : x ∈ H,kxk = 1},where h·,· denotes the inner product in H.In [9] V. Matache determined the shape W(C

In this paper we characterize continuity and compactness of composition operators Cø; mapping the α-Bloch space into the μ-Bloch space, where μ is a weight defined on the unit disk D, in term of certain expression that involve the n-power of the symbol φ.

Tamed spaces – Dirichlet spaces with distribution-valued Ricci bounds

We consider composition operators on Hardy spaces of a half-plane. We mainly study boundedness and compactness. We prove that on these spaces there are no compact composition operators.