Characterization of frame vectors for $ \mathcal{A} $-group-like unitary systems on Hilbert $ C^{\ast} $-modules

Abstract

<abstract><p>In this paper, the notion of an $ \mathcal{A} $-group-like unitary system on a Hilbert $ C^{\ast} $-module is introduced and some basic properties are studied, where $ \mathcal{A} $ is a unital $ C^{\ast} $-algebra. Let $ \mathcal{U} $ be such a unitary system. We prove that a complete Parseval frame vector for $ \mathcal{U} $ can be dilated to a complete wandering vector. Also, it is shown that the set of all the complete Bessel vectors for $ \mathcal{U} $ can be parameterized by the set of the adjointable operators in the double commutant of $ \mathcal{U} $, and that the frame multiplicity of $ \mathcal{U} $ is always finite.</p></abstract>