We study Carleson measures and Toeplitz operators on the class of so-called small weighted Bergman spaces, introduced recently by Seip. A characterization of Carleson measures is obtained which extends Seip’s results from the unit disk of $ℂ$ to the unit ball of ${ℂ}^n$. We use this characterization to give necessary and sufficient conditions for the boundedness and compactness of Toeplitz operators. Finally, we study the Schatten $p$ classes membership of Toeplitz operators for $1<p<\infty$.

We characterize Carleson measures for the analytic Besov spaces. The problem is first reduced to a discrete question involving measures on trees which is then solved. Applications are given to multipliers for the Besov spaces and to the determination of interpolating sequences. The discrete theorem is also applied to analysis of function space on trees.

We study connected components of a common zero set of equivalent singular inner functions in the maximal ideal space of the Banach algebra of bounded analytic functions on the open unit disk. To study topological properties of zero sets of inner functions, we give a new type of factorization theorem for inner functions.

We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the $n$-dimensional complex plane. Characterization of the commutant of such operators is given.

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 ${²}_{\alpha }$.

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.

H. Jiang and C. Lin [Chinese Ann. Math. 23 (2002)] proved that there exist infinitely many Banach spaces, called refined Besov spaces, lying strictly between the Besov spaces $B_{p,q}^s\left(ℝⁿ\right)$ and ${\bigcup }_{t>s}{B}_{p,q}^t\left(ℝⁿ\right)$. In this paper, we prove a similar result for the analytic Besov spaces on the unit disc . We base our construction of the intermediate spaces on operator theory, or, more specifically, the theory of symmetrically normed ideals, introduced by I. Gohberg and M. Krein. At the same time, we use these spaces as models to...

This paper characterizes the Banach algebras of continuous functions on which the spectral factorization mapping 𝔖 is continuous or bounded. It is shown that 𝔖 is continuous if and only if the Riesz projection is bounded on the algebra, and that 𝔖 is bounded only if the algebra is isomorphic to the algebra of continuous functions. Consequently, 𝔖 can never be both continuous and bounded, on any algebra under consideration.

It is well known that the Taylor series of every function in the Fock space $F_{\alpha }^p$ converges in norm when 1 < p < ∞. It is also known that this is no longer true when p = 1. In this note we consider the case 0 < p < 1 and show that the Taylor series of functions in $F_{\alpha }^p$ do not necessarily converge “in norm”.

We first show that the Gaussian integral means of $f:ℂ\to ℂ$ (with respect to the area measure ${\mathrm{e}}^{-{\alpha \left|z\right|}^2}\mathrm{d}A\left(z\right)$) is a convex function of $r$ on $(0,\infty )$ when $\alpha \le 0$. We then prove that the weighted integral means $A_{\alpha ,\beta }(f,r)$ and $L_{\alpha ,\beta }(f,r)$ of the mixed area and the mixed length of $f\left(r𝔻\right)$ and $\partial f\left(r𝔻\right)$, respectively, also have the property of convexity in the case of $\alpha \le 0$. Finally, we show with examples that the range $\alpha \le 0$ is the best possible.