Compact differences of composition operators on Bloch and Lipschitz spaces

Peer reviewe

Generalized Stević-Sharma type operators from derivative Hardy spaces into Zygmund-type spaces

Let $u, v$ be two analytic functions on the open unit disk ${\mathbb D}$ in the complex plane, $\varphi$ an analytic self-map of ${\mathbb D}$, and $m, n$ nonnegative integers such that m < n . In this paper, we consider the generalized Stević-Sharma type operator $T_{u, v, \varphi}^{m, n}f(z) = u(z)f^{(m)}(\varphi(z))+v(z)f^{(n)}(\varphi(z))$ acting from the derivative Hardy spaces into Zygmund-type spaces, and investigate its boundedness, essential norm and compactness

Note on a new class of operators between some spaces of holomorphic functions

The boundedness and compactness of a new class of linear operators from the weighted Bergman space to the weighted-type spaces on the unit ball are characterized

Carleson measures for Besov spaces on the ball with applications

Carleson and vanishing Carleson measures for Besov spaces on the unit ball of C N are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type

Carleson measures for Besov spaces on the ball with applications

Carleson and vanishing Carleson measures for Besov spaces on the unit ball of CN are characterized in terms of Berezin transforms and Bergman-metric balls. The measures are defined via natural imbeddings of Besov spaces into Lebesgue classes by certain combinations of radial derivatives. Membership in Schatten classes of the imbeddings is considered too. Some Carleson measures are not finite, but the results extend and provide new insight to those known for weighted Bergman spaces. Special cases pertain to Arveson and Dirichlet spaces, and a unified view with the usual Hardy-space Carleson measures is presented by letting the order of the radial derivatives tend to 0. Weak convergence in Besov spaces is also characterized, and weakly 0-convergent families are exhibited. Applications are given to separated sequences, operators of Forelli-Rudin type, gap series, characterizations of weighted Bloch, Lipschitz, and growth spaces, inequalities of Fejér-Riesz and Hardy-Littlewood type, and integration operators of Cesàro type. © 2007 Elsevier Inc. All rights reserved

Operator Theory and Harmonic Analysis

The major topics discussed in this workshop were the Feichtinger conjecture and related questions of harmonic analysis, the corona problem for the ball Bn, the weighted approximation problem, and questions related to the model spaces, to multipliers, (hyper-)cyclicity, differentiability, Bezout and Fermat equations, traces and Toeplitz operators in different function spaces. A list of open problems raised at this workshop is also included

Differentiation and composition on the Hardy and Bergman spaces

Banach spaces of analytic functions are defined by norming a collection of these functions defined on a set X. Among the most studied are the Hardy and Bergman spaces of analytic functions on the unit disc in the complex plane. This is likely due to the richness of these spaces. An analytic self-map of the unit disc induces a composition operator on these spaces in the natural way. Beginning with independent papers by E. Nordgren and J. V. Ryff in the 1960\u27s, much work has been done to relate the properties of the composition operator to the characteristics of the inducing map. Every composition operator induced by an analytic self-map of the unit disc is bounded on the Hardy and Bergman spaces. Differentiation is another linear operation which is natural on spaces of analytic functions. Unlike the composition operator, the differentiation operator is poorly behaved on the Hardy and Bergman spaces; that is, it is not a bounded operator. We define a linear operator, possibly unbounded, by applying composition followed by differentiation; that is, for f in a Hardy or Bergman space and an analytic self-map of the disk, $\phi$,DC\sb\phi(f)=(f\circ\phi)\prime.We have found a characterization for the boundedness of this operator on the Hardy space in terms of the inducing map. The operator is bounded exactly when the image of the self-map of the disc is contained in a compact subset of the disc. In contrast, we have found a self-map of the disc with supremum norm equal to one that induces a bounded operator on the Bergman spaces. In this setting we have found conditions necessary for boundedness, and conditions sufficient to imply boundedness. These conditions are closely related. The techniques used involve Carleson-type measures on the unit disc. A very general question arising out of this work involves relating boundedness of the differentiation operator to characteristics of these measures

Thermodynamic laws in isolated systems

The recent experimental realization of exotic matter states in isolated quantum systems and the ensuing controversy about the existence of negative absolute temperatures demand a careful analysis of the conceptual foundations underlying microcanonical thermostatistics. Here, we provide a detailed comparison of the most commonly considered microcanonical entropy definitions, focussing specifically on whether they satisfy or violate the zeroth, first and second law of thermodynamics. Our analysis shows that, for a broad class of systems that includes all standard classical Hamiltonian systems, only the Gibbs volume entropy fulfills all three laws simultaneously. To avoid ambiguities, the discussion is restricted to exact results and analytically tractable examples.Comment: footnotes 19, 26 and outlook section adde