We characterize the membership in the Schatten ideals $\mathcal {S}_p$, $0<p<\infty $, of composition operators acting on weighted Dirichlet spaces. Our results concern a large class of weights. In particular, we examine the case of perturbed superharmonic weights. Characterization of composition operators acting on weighted Bergman spaces to be in $\mathcal {S}_p$ is also given.

We characterize Fredholmness of Toeplitz operators acting on generalized Fock spaces of the n-dimensional complex space for symbols in the space of vanishing mean oscillation VMO. Our results extend the recent characterizations for Toeplitz operators on standard weighted Fock spaces to the setting of generalized weight functions and also allow for unbounded symbols in VMO for the first time. Another novelty is the treatment of small exponents 0 < p < 1, which to our knowledge has not been seen previously in the study of the Fredholm properties of Toeplitz operators on any function spaces. We accomplish this by describing the dual of the generalized Fock spaces with small exponents.

Let $A_{{\it\alpha}}^{p}$ be the weighted Bergman space of the unit ball in ${\mathcal{C}}^{n}$, $n\geq 2$. Recently, Miao studied products of two Toeplitz operators defined on $A_{{\it\alpha}}^{p}$. He proved a necessary condition and a sufficient condition for boundedness of such products in terms of the Berezin transform. We modify the Berezin transform and improve his sufficient condition for products of Toeplitz operators. We also investigate products of two Hankel operators defined on $A_{{\it\alpha}}^{p}$, and products of the Hankel operator and the Toeplitz operator. In particular, in both cases, we prove sufficient conditions for boundedness of the products.

In this paper, we completely determine the commutativity of two Toeplitz operators on the harmonic Bergman space with integrable quasihomogeneous symbols, one of which is of the form $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}e^{ik\theta }r^{\, {m}}$. As an application, the problem of when their product is again a Toeplitz operator is solved. In particular, Toeplitz operators with bounded symbols on the harmonic Bergman space commute with $T_{e^{ik\theta }r^{\, {m}}}$ only in trivial cases, which appears quite different from results on analytic Bergman space in Čučković and Rao [‘Mellin transform, monomial symbols, and commuting Toeplitz operators’, J. Funct. Anal.154 (1998), 195–214].

We consider in this paper the question of when the finite sum of products of two Toeplitz operators is a finite-rank perturbation of a single Toeplitz operator on the Hardy space over the unit disk. A necessary condition is found. As a consequence we obtain a necessary and sufficient condition for the product of three Toeplitz operators to be a finite-rank perturbation of a single Toeplitz operator.

For a given irrational number $\theta$, we define Toeplitz operators with symbols in the irrational rotation algebra ${{\mathcal{A}}_{\theta }}$, and we show that the ${{C}^{*}}$ algebra $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ generated by these Toeplitz operators is an extension of ${{\mathcal{A}}_{\theta }}$ by the algebra of compact operators. We then use these extensions to explicitly exhibit generators of the group $K{{K}^{1}}\left( {{\mathcal{A}}_{\theta }},\mathbb{C} \right)$. We also prove an index theorem for $\mathfrak{J}\left( {{\mathcal{A}}_{\theta }} \right)$ that generalizes the standard index theorem for Toeplitz operators on the circle.

We develop the Fredholm theory for Toeplitz operators, with symbols in the C*-algebra $D = [SO, SAP]_{n, n}$ generated by all slowly oscillating (SO) and semi-almost periodic (SAP) $n\times n$ matrix functions, on the Hardy spaces $H^p_n$ (with $1 < p < \infty$) over the upper half-plane. Using limit operator techniques, we get necessary Fredholm conditions for any operator in the Banach algebra ${\rm alg}(S, D)$ of singular integral operators with coefficients in $D$ on the space $[L^p (\mathbb{R})]_n$. Applying the Allan–Douglas local principle and the theory of Toeplitz operators with SAP matrix symbols, we also establish Fredholm criteria for Toeplitz operators with matrix symbols $g \in D$ on the space $H^p_n$. An index formula for Fredholm Toeplitz operators with matrix symbols in $D$ is obtained on the basis of an appropriate approximation of slowly oscillating components of the symbols.