Here we consider when the difference of two composition operators is compact on the weighted Dirichlet spaces . Specifically we study differences of composition operators on the Dirichlet space and S 2, the space of analytic functions whose first derivative is in H 2, and then use Calderón’s complex interpolation to extend the results to the general weighted Dirichlet spaces. As a corollary we consider composition operators induced by linear fractional self-maps of the disk.

Compact composition operators on $H^{\infty }\left(G\right)$, where G is a region in the complex plane, and the spectra of these operators were described by D. Swanton ( Compact composition operators on B(D), Proc. Amer. Math. Soc. 56 (1976), 152-156). In this short note we characterize all compact endomorphisms, not necessarily those induced by composition operators, on $H^{\infty }\left(D\right)$, where D is the unit disc, and determine their spectra.

We characterize compact composition operators acting on weighted Bergman-Orlicz spaces ${}_{\alpha }^{\psi }=f\in H\left(\right):{\int }_{}\psi \left(\right|f\left(z\right)\left|\right)d{A}_{\alpha }\left(z\right)<\infty$, where α > -1 and ψ is a strictly increasing, subadditive convex function defined on [0,∞) and satisfying ψ(0) = 0, the growth condition $li{m}_{t\to \infty }\psi \left(t\right)/t=\infty$ and the Δ₂-condition. In fact, we prove that $C_{\phi }$ is compact on ${}_{\alpha }^{\psi }$ if and only if it is compact on the weighted Bergman space ${²}_{\alpha }$.

This paper identifies a class of complex symmetric weighted composition operators on $H^2\left(𝔻\right)$ that includes both the unitary and the Hermitian weighted composition operators, as well as a class of normal weighted composition operators identified by Bourdon and Narayan. A characterization of algebraic weighted composition operators with degree no more than two is provided to illustrate that the weight function of a complex symmetric weighted composition operator is not necessarily linear fractional.

We characterize stability under composition of ultradifferentiable classes defined by weight sequences M, by weight functions ω, and, more generally, by weight matrices , and investigate continuity of composition (g,f) ↦ f ∘ g. In addition, we represent the Beurling space ${}^{\left(\omega \right)}$ and the Roumieu space ${}^{\omega }$ as intersection and union of spaces ${}^{\left(M\right)}$ and ${}^M$ for associated weight sequences, respectively.

The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced operator.

We characterize the boundedness and compactness of composition operators from weighted Bergman-Privalov spaces to Zygmund type spaces on the unit disk.

In this work, we begin with a survey of composition operators on the Hardy space H² and on the Wiener algebra A⁺ of absolutely convergent Taylor series, with special emphasis on their compactness, or invertibility, or isometric character. The main results are due respectively to J. Shapiro and D.~Newman. In a second part, we present more recent results, due to Gordon and Hedenmalm on the one hand, and to Bayart, the author et al. on the other hand, concerning the analogues of H² and A⁺ in the setting...

Let $N_{\alpha }$,B and Qβ be the weighted Nevanlinna space, the Bloch space and the Q space, respectively. Note that B and $Q_{\beta }$ are Möbius invariant, but $N_{\alpha }$ is not. We characterize, in function-theoretic terms, when the composition operator $C_{\varphi }f=f◦\varphi$ induced by an analytic self-map ϕ of the unit disk defines an operator $C_{\varphi }:N_{\alpha }\to B$, $B\to Q_{\beta }$, $N_{\alpha }\to Q_{\beta }$ which is bounded resp. compact.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\mathrm{\infty }}$ be a sequence of positive numbers and $1\le p<\mathrm{\infty }$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\mathrm{\infty }}\stackrel{⌃}{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\mathrm{\infty }}{\left|\stackrel{⌃}{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\mathrm{\infty }$ . Suppose that $\frac{1}{p}+\frac{1}{q}=1$ and ${\sum }_{n=1}^{\mathrm{\infty }}\frac{n^{qj}}{\beta {\left(n\right)}^q}=\mathrm{\infty }$ for some nonnegative integer $j$. We show that if $C_{\phi }$ is compact on $H^p\left(\beta \right)$, then the non-tangential limit of ${\phi }^{\left(j+1\right)}$ has modulus greater than one at each boundary point of the open unit disc. Also we show that if $C_{\phi }$ is Fredholm on $H_p\left(\beta \right)$, then $\phi$ must be an automorphism of the open unit disc.

The invertible, closed range, compact, Fredholm and isometric composition operators on Musielak-Orlicz spaces of Bochner type are characterized in the paper.