arXiv subjects
Publications and source records attributed to Gilles Cassier.
The purpose of this paper is to analysis the Harnack part of some truncated shifts whose $ρ$-numerical radius equal one in the finite dimensional case. As pointed out in Theorem 1.17 [12], a key point is to describe the null spaces of the $ρ$-operatorial kernel of these truncated shifts. We establish two fundamental results in this direction and some applications are also given.
The purpose of this paper is to describe the Harnack parts for the operators of class C $ρ$ ($ρ$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with $ρ$-numerical radius one. The case of operators with $ρ$-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact $ρ$-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular attention. Moreover, we study many properties of Harnack equivalence in the general case.
We say that a complex number $λ$ is an extended eigenvalueof a bounded linear operator T on a Hilbert space H if there exists anonzero bounded linear operator X acting on H, called extended eigen-vector associated to $λ$, and satisfying the equation T X = $λ$XT . In thispaper we describe the sets of extended eigenvalues and extended eigen-vectors for the product of a positive and a self-adjoint operator whichare both injective. We also treat the case of normal operators.
Le but de cet article est d'obtenir la cyclicite de certaines classes de fonctions pour des operateurs de Toeplitz anti-analytique associes a un produit fini de Blaschke dans les espaces $H^p$ ou $1<p<\infty$. Il s'agit aussi de decrire les sous-espaces invariants par ce type d'operateur et engendres par des decompositions lacunaires de fonctions.
An equivalent formulation of the von Neumann inequality states that the backward shift $S^*$ on $\ell_{2}$ is extremal, in the sense that if $T$ is a Hilbert space contraction, then $\|p(T)\| \leq \|p(S^*)\|$ for each polynomial $p$. We discuss several results of the following type : if $T$ is a Hilbert space contraction satisfying some constraints, then $S^*$ restricted to a suitable invariant subspace is an extremal operator. Several operator radii are used instead of the operator norm. Applications to inequalities of coefficients of rational functions positive on the torus are given.
Let $M$ be a finite von Neumann algebra. In the first part, we give asymptotic results about $M$-stable sequences of weak*-continuous mappings which are related with operators belonging to $M$. In the second part, we extend, by a shorter way, similarity results given in [CaFa2] to unbounded semigroups of operators contained in a finite von Neumann algebra.