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.

Let Ω ⊂ ℝn be an open set and X(Ω) be any rearrangement invariant function space close to L q(Ω), i.e. X has the q-scaling property. We prove that each homeomorphism f which induces the composition operator u ↦ u ℴ f from W 1 X to W 1 X is necessarily a q-quasiconformal mapping. We also give some new results for the sufficiency of this condition for the composition operator.

This paper studies the compression of a $k$th-order slant Toeplitz operator on the Hardy space $H^2\left({𝕋}^n\right)$ for integers $k\ge 2$ and $n\ge 1$. It also provides a characterization of the compression of a $k$th-order slant Toeplitz operator on $H^2\left({𝕋}^n\right)$. Finally, the paper highlights certain properties, namely isometry, eigenvalues, eigenvectors, spectrum and spectral radius of the compression of $k$th-order slant Toeplitz operator on the Hardy space $H^2\left({𝕋}^n\right)$ of $n$-dimensional torus ${𝕋}^n$.

Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined...