A bounded linear operator T on a Banach space X is called an (m,p)-isometry for a positive integer m and a real number p ≥ 1 if, for any vector x ∈ X, ${\sum }_{k=0}^m{(-1)}^k\left(\genfrac{}{}{0pt}{}{m}{k}\right)\left|\right|T^{k}x{\left|\right|}^p=0$. We prove that any power of an (m,p)-isometry is also an (m,p)-isometry. In general the converse is not true. However, we prove that if $T^r$ and $T^{r+1}$ are (m,p)-isometries for a positive integer r, then T is an (m,p)-isometry. More precisely, if $T^r$ is an (m,p)-isometry and $T^s$ is an (l,p)-isometry, then $T^t$ is an (h,p)-isometry, where t = gcd(r,s) and h = min(m,l)....

The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and...

Dans un exposé précédent [1], nous avons justifié l’introduction de l’équation de Szegö cubique comme cas modèle d’équation de type Schrödinger sans dispersion. Ce cas modèle s’est révélé être intéressant sous divers aspects [2]. Dans cet exposé, nous nous attacherons à montrer comment la complète intégrabilité de l’équation de Szegö cubique permet de résoudre un problème spectral inverse pour les opérateurs de Hankel.

We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators $T_f{T}_g$ on the harmonic Bergman space is equal to a Toeplitz operator $T_h$, then the product $T_g{T}_f$ is also the Toeplitz operator $T_h$, and hence $T_f$ commutes with $T_g$. From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator.

Let V be the C*-algebra B(H) of bounded linear operators acting on the Hilbert space H, or the Jordan algebra S(H) of self-adjoint operators in B(H). For a fixed sequence (i₁, ..., iₘ) with i₁, ..., iₘ ∈ 1, ..., k, define a product of $A₁,...,A_k\in V$ by $A₁*\cdots *A_k=A_{i₁}\cdots A_{iₘ}$. This includes the usual product $A₁*\cdots *A_k=A₁\cdots A_k$ and the Jordan triple product A*B = ABA as special cases. Denote the numerical range of A ∈ V by W(A) = (Ax,x): x ∈ H, (x,x) = 1. If there is a unitary operator U and a scalar μ satisfying ${\mu }^m=1$ such that ϕ: V → V has the form A...

We first determine when the sum of products of Hankel and Toeplitz operators is equal to zero; then we characterize when the product of a Toeplitz operator and a Hankel operator is a compact perturbation of a Hankel operator or a Toeplitz operator and when it is a finite rank perturbation of a Toeplitz operator.

Soient $A$ une $C^{*}$-algèbre approximativement finie simple avec unité, $G{L}_1\left(A\right)$ le groupe des inversibles et $U_1\left(A\right)$ le groupe des unitaires de $A$. Nous avons défini dans un précédent travail un homomorphisme ${\Delta }_T$, appelé déterminant universel de $A$, de $G{L}_1\left(A\right)$ sur un groupe abélien associé à $A$. Nous montrons ici que, pour qu’un élément $x$ dans $G{L}_1\left(A\right)$ ou dans $U_1\left(A\right)$ soit produit d’un nombre fini de commutateurs, il (faut et il) suffit que $x\in \mathrm{Ker}\left({\Delta }_T\right).$ Ceci permet en particulier d’identifier le noyau de la projection canonique $K_1\left(A\right)\to K_1^{\mathrm{top}}\left(A\right).$ On établit aussi...