Characterization of $K$-frame vectors and $K$-frame generator multipliers

Abstract

‎‎Let $\mathcal{U}$ be a unitary system and let $\mathcal{B(U)}$ be the Bessel vector space for $\mathcal{U}$‎. ‎In this article‎, ‎we give a characterization of Bessel vector spaces and local commutant spaces at different complete frame vectors‎. ‎The relation between local commutant spaces at different complete frame vectors is investigated‎. ‎Moreover‎, ‎by introducing multiplication and adjoint on the Bessel vector space for a unital semigroup of unitary operators‎, ‎we give a $C^*$-algebra structure to $\mathcal{B(U)}$‎. ‎Then‎, ‎we construct some subsets of $K$-frame vectors that have a Banach space or Banach algebra structure‎. ‎Also‎, ‎as a consequence‎, ‎the set of complete frame vectors for different unitary systems contains Banach spaces or Banach algebras‎. ‎In the end‎, ‎we give several characterizations of $K$-frame generator multipliers and Parseval $K$-frame generator multipliers‎.