Integral operators induced by symbols with non-negative Maclaurin coefficients mapping into $H^\infty$

Uniform approximation of Bloch functions and the boundedness of the integration operator on $H^\infty$

We obtain a necessary and sufficient condition for the operator of integration to be bounded on $H^\infty$ in a simply connected domain. The main ingredient of the proof is a new result on uniform approximation of Bloch functions. This gives a full characterization of symbols of certain Volterra operators that act on bounded analytic functions in the disc if the symbol is assumed to be univalent. Without this assumption the answer is not known, and as the example at the end of the paper shows, the natural answer is definitely false.

作用于 $H^{\infty}$ 的一些积分算子

Some integral operators acting on $H^{\infty}$

本文研究了积分算子 $T_g$ 及其伴随算子 $S_g$ 作用于有界解析函数空间 $H^\infty$ 时的有界性与紧致性，刻画了相应符号 $g$ 的条件，并探讨了符号为单叶时的单叶函数理论问题。

Asymmetric truncated Toeplitz operators and Toeplitz operators with matrix symbol

Truncated Toeplitz operators and their asymmetric versions are studied in the context of the Hardy space $H^p$ of the half-plane for $1<p<\infty$. It is shown that they are equivalent after extension to $2 \times 2$ matricial Toeplitz operators, which allows one to deduce information about their invertibility properties. Shifted model spaces are presented in the context of invariant subspaces, allowing one to deduce new Beurling--Lax theorems.

The equivalence of pseudodifferential operators and their symbols via Čech-Dolbeault cohomology

In this paper we construct the sheaf morphism from the sheaf of pseudodifferential operators to its symbol class. Since the map is hard to construct directly, we realize it with two original ideas as follows. First, to calculate cohomologies we use the theory of Čech-Dolbeault cohomology introduced by Honda, Izawa and Suwa. Secondly we construct a new symbol class, which is called the symbols of $C^\infty$-type. These ideas enable us to construct the sheaf morphism, which is actually an isomorphism of sheaves.

Mean Ergodic Composition Operators on Banach spaces of holomorphic functions

Given a symbol $φ,$ i.e., a holomorphic endomorphism of the unit disc, we consider the composition operator $C_φ(f)=f\circφ$ defined on the Banach spaces of holomorphic functions $A(\mathbb{D})$ and $H^{\infty}(\mathbb{D})$. We obtain different conditions on the symbol $φ$ which characterize when the composition operator is mean ergodic and uniformly mean ergodic in the corresponding spaces. These conditions are related to the asymptotic behaviour of the iterates of the symbol. As an appendix, we deal with some particular case in the setting of weighted Banach spaces of holomorphic functions.

Partially isometric Toeplitz operators on the polydisc

A Toeplitz operator $T_φ$, $φ\in L^\infty(\mathbb{T}^n)$, is a partial isometry if and only if there exist inner functions $φ_1, φ_2 \in H^\infty(\mathbb{D}^n)$ such that $φ_1$ and $φ_2$ depends on different variables and $φ= \barφ_1 φ_2$. In particular, for $n=1$, along with new proof, this recovers a classical theorem of Brown and Douglas. \noindent We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in $H^\infty(\mathbb{D}^n)$. Moreover, partially isometric Toeplitz operators are always power partial isometry …

具有倍权函数的Bergman空间上的Toeplitz和Hankel算子

Toeplitz and Hankel operators on Bergman spaces with doubling weights

本文研究倍权Bergman空间上Toeplitz与Hankel算子的有界性和紧性，通过Berezin变换刻画有界性，并给出共解析符号情形下的充要条件。

Boundedness and Compactness of Toeplitz operators with L^1 symbols on the Bergman space

We characterise the boundedness of a Toeplitz operator on the Bergman space with an L^1 symbol.We also prove that the compactness of a Toeplitz operator on the Bergman space with an L^1 symbol is completely determined by the boundary behaviour of itss Berezin transform. This result extends known results in the cases when the symbol is either a positive L^1 function, an L^\infty function or a general BMO^1 function.

Compactness of operators on generalized Fock spaces

For a very general class of weighted Fock spaces on $\mathbb{C}^n$, we give necessary and sufficient conditions for a Toeplitz operator with a (not necessarily positive) measure symbol to be compact. Furthermore, we show that all compact operators are in the norm closure of the algebra generated by Toeplitz operators with $C_c ^\infty(\mathbb{C}^n)$ symbols, and in the Hilbert space setting show that all compact operators are in the norm closure of the set of such Toeplitz operators.

Multichannel scattering theory for Toeplitz operators with piecewise continuous symbols

Self-adjoint Toeplitz operators have purely absolutely continuous spectrum. For Toeplitz operators $T$ with piecewise continuous symbols, we suggest a further spectral classification determined by propagation properties of the operator $T$, that is, by the behavior of $\exp(-iTt) f$ for $t\to\pm\infty$. It turns out that the spectrum is naturally partitioned into three disjoint subsets: thick, thin and mixed spectra. On the thick spectrum, the propagation properties are modeled by the continuous part of the symbol, whereas on the thin spectrum, the model operator is determined by the jumps of …

由可测符号诱导的作用于加权Bergman空间的小Hankel算子

Small Hankel operator induced by measurable symbol acting on weighted Bergman spaces

本文在径向权类$\mathcal{D}$上完整刻画了由可测符号诱导的小Hankel算子从加权Bergman空间到其共轭解析空间的有界性，并建立了修正Bergman核的精确积分估计。

Localization and compactness of Operators on Fock Spaces

For $0<p\leq\infty$, let $F^{p}_φ$ be the Fock space induced by a weight function $φ$ satisfying $ dd^c φ\simeq ω_0$. In this paper, given $p\in (0, 1]$ we introduce the concept of weakly localized operators on $ F^{p}_φ$, we characterize the compact operators in the algebra generated by weakly localized operators. As an application, for $0<p<\infty$ we prove that an operator $T$ in the algebra generated by bounded Toeplitz operators with $\textrm{BMO}$ symbols is compact on $F^p_φ$ if and only if its Berezin transform satisfies certain vanishing property at $\infty$. In the classical Fock spa…

Fredholmness and compactness of truncated Toeplitz and Hankel operators

We prove the spectral mapping theorem $σ_e(A_ϕ) = ϕ(σ_e(A_z))$ for the Fredholm spectrum of a truncated Toeplitz operator $A_ϕ$ with symbol $ϕ$ in the Sarason algebra $C+H^\infty$ acting on a coinvariant subspace $K_θ$ of the Hardy space $H^2$. Our second result says that a truncated Hankel operator on the subspace $K_θ$ generated by a one-component inner function $θ$ is compact if and only if it has a continuous symbol. We also suppose a description of truncated Toeplitz and Hankel operators in Schatten classes $S^p$.