### 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.

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 3 (2019), 587-603.

Dates
Accepted: 18 December 2018
First available in Project Euclid: 2 March 2019

https://projecteuclid.org/euclid.aot/1551495622

Digital Object Identifier
doi:10.15352/aot.1808-1408

Mathematical Reviews number (MathSciNet)
MR3919033

Zentralblatt MATH identifier
07056787

Subjects
Primary: 42C15: General harmonic expansions, frames
Secondary: 42A38‎ ‎41A58

#### Citation

Javani, Somayeh; Takhteh, Farkhondeh. Characterization of $K$-frame vectors and $K$-frame generator multipliers. Adv. Oper. Theory 4 (2019), no. 3, 587--603. doi:10.15352/aot.1808-1408. https://projecteuclid.org/euclid.aot/1551495622

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