UNITARY SKEW-DILATIONS OF HILBERT SPACE OPERATORS

Vidal Agniel (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

The aim of this paper is to study, for a given sequence $(\rho_n)_{n\ge1}$ of complex numbers, the class of Hilbert space operators possessing $(\rho_n)$-unitary dilations. This is the class of bounded linear operators $T$ acting on a Hilbert space $H$, whose iterates $T^n$ can be represented as $T^n=\rho_n P_H U^n|_H$, $n \ge 1$, for some unitary operator $U$ acting on a larger Hilbert space, containing $H$ as a closed subspace. Here $P_H$ is the projection from this larger space onto $H$. The case when all $\rho_n$'s are equal to a positive real number $\rho$ leads to the class $C_{\rho}$ introduced in the 1960s by Foias and Sz.-Nagy, while the case when all $\rho_n$'s are positive real numbers has been previously considered by several authors. Some applications and examples of operators possessing $(\rho_n)$-unitary dilations, showing a behavior different from the classical case, are given in this paper.

Domaines: Mathématiques [math]; Analyse fonctionnelle [math.FA]; Algèbres d'opérateurs [math.OA]

Vidal Agniel. UNITARY SKEW-DILATIONS OF HILBERT SPACE OPERATORS. Extracta Mathematica, 2020, 35 (2). ⟨hal-02570997⟩