The Hilbert matrix acts on Hardy spaces by multiplication with Taylor coefficients. We find an upper bound for the norm of the induced 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 ${𝕋}^n$.

For a locally compact group G we consider the algebra CD(G) of convolution-dominated operators on L²(G), where an operator A: L²(G) → L²(G) is called convolution-dominated if there exists a ∈ L¹(G) such that for all f ∈ L²(G) |Af(x)| ≤ a⋆|f|(x), for almost all x ∈ G. (1) The case of discrete groups was treated in previous publications [fgl08a, fgl08]. For non-discrete groups we investigate a subalgebra of regular convolution-dominated operators generated by product convolution operators, where the...

It is well known that one can often construct a star-product by expanding the product of two Toeplitz operators asymptotically into a series of other Toeplitz operators multiplied by increasing powers of the Planck constant h. This is the Berezin-Toeplitz quantization. We show that one can obtain in a similar way in fact any star-product which is equivalent to the Berezin-Toeplitz star-product, by using instead of Toeplitz operators other suitable mappings from compactly supported smooth functions...

The discrete Wiener-Hopf operator generated by a function $a\left(e^{i\theta }\right)$ with the Fourier series ${\sum }_{n\in \mathbb{Z}}{a}_n{e}^{in\theta }$ is the operator T(a) induced by the Toeplitz matrix ${\left(a_{j-k}\right)}_{j,k=0}^{\infty }$ on some weighted sequence space $l^p({\mathbb{Z}}_{+},w)$. We assume that w satisfies the Muckenhoupt $A_p$ condition and that a is a piecewise continuous function subject to some natural multiplier condition. The last condition is in particular satisfied if a is of bounded variation. Our main result is a Fredholm criterion and an index formula for T(a). It implies that the essential spectrum...

We give several new characterizations of the dual of the dyadic Hardy space H1,d(T2), the so-called dyadic BMO space in two variables and denoted BMOdprod. These include characterizations in terms of Haar multipliers, in terms of the "symmetrised paraproduct" Λb, in terms of the rectangular BMO norms of the iterated "sweeps", and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between BMOdprod and John-Nirenberg type inequalities, and study a scale...

A gap in the proof of [4, Theorem 1] is removed.

Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes $\overline{f}$ tels que l’opérateur de Hankel $H_{\overline{f}}$ sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten ${𝒮}^p$ quand $p$ tend vers $1$ et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs. Abstract....

In 1999 Nina Zorboska and in 2003 P. S. Bourdon, D. Levi, S. K. Narayan and J. H. Shapiro investigated the essentially normal composition operator $C_{\varphi }$, when $\varphi$ is a linear-fractional self-map of $𝔻$. In this paper first, we investigate the essential normality problem for the operator $T_w{C}_{\varphi }$ on the Hardy space $H^2$, where $w$ is a bounded measurable function on $\partial 𝔻$ which is continuous at each point of $F\left(\varphi \right)$, $\varphi \in 𝒮\left(2\right)$, and $T_w$ is the Toeplitz operator with symbol $w$. Then we use these results and characterize the essentially normal...

Using a factorization lemma we obtain improvements and simplifications of results on representation of generalized Toeplitz and Hankel operators as compression of symbols.

Ortega-Cerdà-Seip demonstrated that there are bounded multiplicative Hankel forms which do not arise from bounded symbols. On the other hand, when such a form is in the Hilbert-Schmidt class ₂, Helson showed that it has a bounded symbol. The present work investigates forms belonging to the Schatten classes between these two cases. It is shown that for every $p>{(1-log\pi /log4)}^{-1}$ there exist multiplicative Hankel forms in the Schatten class ${}_p$ which lack bounded symbols. The lower bound on p is in a certain sense optimal...