We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

For a bounded analytic function $\varphi $ on the unit disk $\mathbb {D}$ with $\|\varphi \|_\infty \le 1$, we consider the defect operators $D_\varphi $ and $D_{\overline \varphi }$ of the Toeplitz operators $T_{\overline \varphi }$ and $T_\varphi $, respectively, on the weighted Bergman space $A^2_\alpha $. The ranges of $D_\varphi $ and $D_{\overline \varphi }$, written as $H(\varphi )$ and $H(\overline \varphi )$ and equipped with appropriate inner products, are called sub-Bergman spaces.

We prove the following three results in the paper: for $-1<\alpha \le 0$, the space $H(\varphi )$ has a complete Nevanlinna–Pick kernel if and only if $\varphi $ is a Möbius map; for $\alpha>-1$, we have $H(\varphi )=H(\overline \varphi )=A^2_{\alpha -1}$ if and only if the defect operators $D_\varphi $ and $D_{\overline \varphi }$ are compact; and for $\alpha>-1$, we have $D^2_\varphi (A^2_\alpha )= D^2_{\overline \varphi }(A^2_\alpha )=A^2_{\alpha -2}$ if and only if $\varphi $ is a finite Blaschke product. In some sense, our restrictions on $\alpha $ here are best possible.

Let $\Omega $ be a bounded Reinhardt domain in $\mathbb {C}^n$ and $\phi _1,\ldots ,\phi _m$ be finite sums of bounded quasi-homogeneous functions. We show that if the product of Toeplitz operators $T_{\phi _m}\cdots T_{\phi _1}=0$ on the Bergman space on $\Omega $ , then $\phi _j=0$ for some j.

This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.

We study the boundedness of Toeplitz operators with locally integrable symbols on Bergman spaces Ap(Ω), 1 < p < ∞, where Ω ⊂ ℂ is a bounded simply connected domain with polygonal boundary. We give sufficient conditions for the boundedness of generalized Toeplitz operators in terms of ‘averages’ of symbol over certain Cartesian squares. We use the Whitney decomposition of Ω in the proof. We also give examples of bounded Toeplitz operators on Ap(Ω) in the case where polygon Ω has such a large corner that the Bergman projection is unbounded.

In this paper, we completely characterize the finite rank commutator and semi-commutator of two monomial-type Toeplitz operators on the Bergman space of certain weakly pseudoconvex domains. Somewhat surprisingly, there are not only plenty of commuting monomial-type Toeplitz operators but also non-trivial semi-commuting monomial-type Toeplitz operators. Our results are new even for the unit ball.

In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.

We formulate and prove a “winding number” index theorem for certain “Toeplitz” operators in the same spirit as Gohberg–Krein, Lesch and others. The “number” is replaced by a self-adjoint operator in a subalgebra $Z\,\subseteq \,Z\left( A \right)$ of a unital $C*$-algebra, $A$. We assume a faithful $Z$-valued trace $\tau $ on $A$ left invariant under an action $\alpha :\,\text{R}\,\to \,\text{Aut}\left( A \right)$ leaving $Z$ pointwise fixed. If $\delta $ is the infinitesimal generator of $\alpha $ and $u$ is invertible in $\text{dom}\left( \delta \right)$, then the “winding operator” of $u$ is $\frac{1}{2\pi i}\tau \left( \delta \left( u \right){{u}^{-1}} \right)\,\in \,{{Z}_{sa}}$. By a careful choice of representations we extend $\left( A,\,Z,\,\tau ,\,\alpha \right)$ to a von Neumann setting $\left( \mathfrak{A},\,\mathfrak{Z},\,\bar{\tau },\,\bar{\alpha } \right)$ where $\mathfrak{A}\,=\,{A}''$ and $\mathfrak{Z}\,=\,{Z}''$. Then $A\,\subset \,\mathfrak{A}\,\subset \,\mathfrak{A}\,\rtimes \,\mathbf{R}$, the von Neumann crossed product, and there is a faithful, dual $\mathfrak{Z}$-trace on $\mathfrak{A}\,\rtimes \,\mathbf{R}$. If $P$ is the projection in $\mathfrak{A}\,\rtimes \,\mathbf{R}$ corresponding to the non-negative spectrum of the generator of $\mathbf{R}$ inside $\mathfrak{A}\,\rtimes \,\mathbf{R}$ and $\tilde{\pi }:\,A\,\to \,\mathfrak{A}\,\rtimes \,\mathbf{R}$ is the embedding, then we define ${{T}_{u}}\,=\,P\tilde{\pi }\left( u \right)P\,\text{for}\,u\,\in \,{{A}^{-1}}$ and show it is Fredholm in an appropriate sense and the $\mathfrak{Z}$-valued index of ${{T}_{u}}$ is the negative of the winding operator. In outline the proof follows that of the scalar case done previously by the authors. The main difficulty is making sense of the constructions with the scalars replaced by $\mathfrak{Z}$ in the von Neumann setting. The construction of the dual $\mathfrak{Z}$-trace on $\mathfrak{A}\,\rtimes \,\mathbf{R}$ requires the nontrivial development of a $\mathfrak{Z}$-Hilbert algebra theory. We show that certain of these Fredholm operators fiber as a “section” of Fredholm operators with scalar-valued index and the centre-valued index fibers as a section of the scalar-valued indices.

Given a positive Borel measure ${\it\mu}$ on the $n$-dimensional Euclidean space $\mathbb{C}^{n}$, we characterise the boundedness (and compactness) of Toeplitz operators $T_{{\it\mu}}$ between Fock spaces $F^{\infty }({\it\varphi})$ and $F^{p}({\it\varphi})$ with $0<p\leq \infty$ in terms of $t$-Berezin transforms and averaging functions of ${\it\mu}$. Our result extends recent work of Mengestie [‘On Toeplitz operators between Fock spaces’, Integral Equations Operator Theory78 (2014), 213–224] and others.

Zelditch's proof of the Tian–Yau–Zelditch Theorem makes use of the Boutet de Monvel–Sjöstrand results on the asymptotic properties of Szegö projectors for strictly pseudoconvex domains. However, as we will show below, the theorem is also closely related to another theorem of Boutet de Monvel's, namely his wave trace formula for Toeplitz operators. Finally, we will derive, for the pseudoconvex manifolds considered by Zelditch in his proof of the Tian–Yau–Zelditch Theorem, an analogue of another result of Boutet de Monvel's, the extendability theorem of Berndtsson for holomorphic functions on Grauert tubes.

We study the boundedness and compactness of Toeplitz operators Ta on Bergman spaces , 1 < p < ∞. The novelty is that we allow distributional symbols. It turns out that the belonging of the symbol to a weighted Sobolev space of negative order is sufficient for the boundedness of Ta. We show the natural relation of the hyperbolic geometry of the disc and the order of the distribution. A corresponding sufficient condition for the compactness is also derived.

We consider the problem of determining for which square integrable functions $f$ and $g$ on the polydisk the densely defined Hankel product ${{H}_{f}}\,H_{g}^{*}$ is bounded on the Bergman space of the polydisk. Furthermore, we obtain similar results for the mixed Haplitz products ${{H}_{g}}\,{{T}_{{\bar{f}}}}$ and ${{T}_{f}}\,H_{g}^{*}$, where $f$ and $g$ are square integrable on the polydisk and $f$ is analytic.

The spectra of the Toeplitz operators on the weighted Hardy space ${{H}^{2}}\,(W\,d\theta \,/\,2\pi )$ and the Hardy space ${{H}^{p}}\,(d\theta \,/\,2\pi )$, and the singular integral operators on the Lebesgue space ${{L}^{2}}\,(d\theta \,/\,2\pi )$ are studied. For example, the theorems of Brown-Halmos type and Hartman-Wintner type are studied.

Suppose λ is an isolated eigenvalue of the (bounded linear) operator T on the Banach space X and the algebraic multiplicity of λ is finite. Let Tn be a sequence of operators on X that converge to T pointwise, that is, Tnx → Tx for every x ∈ X. If ‖(T − Tn)Tn‖ and ‖Tn(T − Tn)‖ converge to 0 then Tn is strongly stable at λ.

In this article we give a sufficient condition for quasisimilar analytic Toeplitz operators to be unitarily equivalent. We also use a result of Deddens and Wong to give a sufficient condition for an operator intertwining two analytic Toeplitz operators to intertwine their inner parts too. Analytic Toeplitz operators with univalent symbols satisfying a suitable normalization that are quasisimilar are shown to have equal symbols.

Let S be a subnormal operator and let be the weak-star closed algebra generated by S and 1. An example of an irreducible cyclic subnormal operator S is found such that there is a T in with S and T quasisimilar but not unitarily equivalent. However, if S is the unilateral shift, T ∈ and S and T are quasisimilar, then S ≅ T.