Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $$\ell ^p$$ ℓ p spaces $$1<p<+\infty $$ 1 < p < + ∞ . Our main result is that when an analytic symbol g is a multiplier for a weighted $$\ell ^p$$ ℓ p space, then the corresponding generalized Volterra operator $$T_g$$ T g is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $$\{0\}.$$ { 0 } . This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of $$\ell ^p$$ ℓ p spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $$\ell ^p$$ ℓ p . We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on $$\ell ^p, 1<p<\infty $$ ℓ p , 1 < p < ∞ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $$\ell ^2 $$ ℓ 2 , extending a result of E. Ricard.

Amenable dynamical systems over locally compact groups

We establish several new characterizations of amenable $W^*$ - and $C^*$ -dynamical systems over arbitrary locally compact groups. In the $W^*$ -setting we show that amenability is equivalent to (1) a Reiter property and (2) the existence of a certain net of completely positive Herz–Schur multipliers of $(M,G,\alpha )$ converging point weak* to the identity of $G\bar {\ltimes }M$ . In the $C^*$ -setting, we prove that amenability of $(A,G,\alpha )$ is equivalent to an analogous Herz–Schur multiplier approximation of the identity of the reduced crossed product $G\ltimes A$ , as well as a particular case of the positive weak approximation property of Bédos and Conti [On discrete twisted $C^*$ -dynamical systems, Hilbert $C^*$ -modules and regularity. Münster J. Math.5 (2012), 183–208] (generalized to the locally compact setting). When $Z(A^{**})=Z(A)^{**}$ , it follows that amenability is equivalent to the1-positive approximation property of Exel and Ng [Approximation property of $C^*$ -algebraic bundles. Math. Proc. Cambridge Philos. Soc.132(3) (2002), 509–522]. In particular, when $A=C_0(X)$ is commutative, amenability of $(C_0(X),G,\alpha )$ coincides with topological amenability of the G-space $(G,X)$ .

On a Polyanalytic Approach to Noncommutative de Branges–Rovnyak Spaces and Schur Analysis

In this paper we begin the study of Schur analysis and of de Branges–Rovnyak spaces in the framework of Fueter hyperholomorphic functions. The difference with other approaches is that we consider the class of functions spanned by Appell-like polynomials. This approach is very efficient from various points of view, for example in operator theory, and allows us to make connections with the recently developed theory of slice polyanalytic functions. We tackle a number of problems: we describe a Hardy space, Schur multipliers and related results. We also discuss Blaschke functions, Herglotz multipliers and their associated kernels and Hilbert spaces. Finally, we consider the counterpart of the half-space case, and the corresponding Hardy space, Schur multipliers and Carathéodory multipliers.

Computing the nonabelian tensor squares of groups of order $$\pmb {p^3q}$$

In this paper, we describe the explicit structures of nonabelian tensor squares of nonabelian groups of order $$p^3q$$ p 3 q , where p and q are distinct prime numbers and $$p>2$$ p > 2 . Our method is based on determining the structures of their Schur multipliers by applying some well-known results and the presentations of groups, which leads us to obtain the orders of their nonabelian tensor squares.