The Nevanlinna algebras, ${}_{\alpha }^p$, of this paper are the $L^p$ variants of classical weighted area Nevanlinna classes of analytic functions on = z ∈ ℂ: |z| < 1. They are F-algebras, neither locally bounded nor locally convex, with a rich duality structure. For s = (α+2)/p, the algebra $F_s$ of analytic functions f: → ℂ such that ${(1-|z\left|\right)}^s\left|f\left(z\right)\right|\to 0$ as |z| → 1 is the Fréchet envelope of ${}_{\alpha }^p$. The corresponding algebra ${}_s^{\infty }$ of analytic f: → ℂ such that $su{p}_{z\in }{(1-|z\left|\right)}^s\left|f\left(z\right)\right|<\infty$ is a complete metric space but fails to be a topological vector space. $F_s$ is also...

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^n$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately.

Let ψ and φ be analytic functions on the open unit disk $𝔻$ with φ($𝔻$) ⊆ $𝔻$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $𝒟$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations...

Let $X$ be a Banach space of analytic functions on the open unit disk and $\Gamma$ a subset of linear isometries on $X$. Sufficient conditions are given for non-supercyclicity of $\Gamma$. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space $H^p$ or the Bergman space $L_a^p$ ($1<p<\infty$, $p\ne 2$) are not supercyclic....

The aim of the paper is to propose a definition of numerical range of an operator on reflexive Banach spaces. Under this definition the numerical range will possess the basic properties of a canonical numerical range. We will determine necessary and sufficient conditions under which the numerical range of a composition operator on a weighted Hardy space is closed. We will also give some necessary conditions to show that when the closure of the numerical range of a composition operator on a small...

This paper is a short survey on the numerical range of some composition operators. The first part is devoted to composition operators on the Hilbert Hardy space H2 on the unit disk. The results are due to P. Bourdon, J. Shapiro and V. Matache.In the second part we study the numerical range of composition operators on the Hilbert space H2 of Dirichlet series. These results are due to H. Queffélec and the author.The third part is devoted to compactness connected with fixed points in the setting of...

The Cauchy dual operator T’, given by $T{(T*T)}^{-1}$, provides a bounded unitary invariant for a closed left-invertible T. Hence, in some special cases, problems in the theory of unbounded Hilbert space operators can be related to similar problems in the theory of bounded Hilbert space operators. In particular, for a closed expansive T with finite-dimensional cokernel, it is shown that T admits the Cowen-Douglas decomposition if and only if T’ admits the Wold-type decomposition (see Definitions 1.1 and 1.2 below)....

Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of $\ell {₂}^{\infty }$ then T can be written as a weighted composition...

Let $\varphi$ be an analytic self-mapping of $𝔻$ and $g$ an analytic function on $𝔻$. In this paper we characterize the bounded and compact Volterra composition operators from the Bergman-type space to the Bloch-type space. We also obtain an asymptotical expression of the essential norm of these operators in terms of the symbols $g$ and $\varphi$.

Let ϕ: → and ψ: → ℂ be analytic maps. They induce a weighted composition operator $\psi C_{\varphi }$ acting between weighted Bergman spaces of infinite order and weighted Bloch type spaces. Under some assumptions on the weights we give a characterization for such an operator to be bounded in terms of the weights involved as well as the functions ψ and ϕ

Description of multiplication operators generated by a sequence and composition operators induced by a partition on Lorentz sequence spaces $l(p,q)$, $1<p\le \infty$, $1\le q\le \infty$ is presented.

We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces $H^p$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with $X=H^p$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into $L_h$. We give a complete characterization...

We prove that a K-quasiconformal mapping f:ℝ² → ℝ² which maps the unit disk onto itself preserves the space EXP() of exponentially integrable functions over , in the sense that u ∈ EXP() if and only if $u\circ f^{-1}\in EXP\left(\right)$. Moreover, if f is assumed to be conformal outside the unit disk and principal, we provide the estimate $1/(1+KlogK)\le \left(\right||u\circ f^{-1}{\left|\right|}_{EXP\left(\right)}{)/(\left|\right|u\left|\right|}_{EXP\left(\right)})\le 1+KlogK$ for every u ∈ EXP(). Similarly, we consider the distance from $L^{\infty }$ in EXP and we prove that if f: Ω → Ω’ is a K-quasiconformal mapping and G ⊂ ⊂ Ω, then $1/K\le \left(dis{t}_{EXP\left(f\right(G\left)\right)}(u\circ f^{-1},L^{\infty }\left(f\left(G\right)\right))\right)/\left(dis{t}_{EXP\left(f\right(G\left)\right)}(u,L^{\infty }\left(G\right))\right)\le K$ for every u ∈ EXP(). We also prove that...