# Journal of Operator Theory

Volume 49, Issue 2, Spring 2003 pp. 223-244.

Frame representations for group-like unitary operator systems**Authors**: Jean-Pierre Gabardo (1) and Deguang Han (2)

**Author institution:**(1) Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada

(2) Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA

**Summary:**A group-like unitary system ${\cal U}$ is a set of unitary operators such that the group generated by the system is contained in ${\bbb T}{\cal U}$, where ${\bbb T}$ denotes the unit circle. Every frame representation for a group-like unitary system is (unitarily equivalent to) a subrepresentation of its left regular representation and the norm of a normalized tight frame vector determines the redundancy of the representation. In the case that a group-like unitary system admits enough Bessel vectors, the commutant of the system can be characterized in terms of the analysis operators associated with all the Bessel vectors. This allows us to define a natural quantity (the frame redundancy) for the system which will determine when the system admits a cyclic vector. A simple application of this leads to an elementary proof to the well-known time-frequency density theorem in Gabor analysis.

**Keywords:**Group-like unitary systems, Gabor systems, frame vectors, von Neumann algebras, frame representations, analysis operators

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