Frames and generalized operator orbits

Ole Christensen, Marzieh Hasannasab*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

This short note is concerned with various operator representations of frames in a Hilbert space H. While it is known that only very special frames can be represented in the form {Tnφ}∞/n=0 for a bounded operator T, we prove that every frame (actually, every Bessel sequence) has a representation of the form {UTkφ}∞/k=0 for certain bounded operators U and T. We also provide a lifting procedure that allows to represent any given Bessel sequence in the form {PTkφ}∞/k=0, where T is a bounded operator on an ambient Hilbert space and P denotes the orthogonal projection onto the given Hilbert space H. In particular, this implies that for any frame, the frame coefficients of f∈ H can be calculated as inner products between ƒ and a system of functions of the form {Tkφ}∞/k=0, for a bounded operator T on an ambient Hilbert space.

Original languageEnglish
Article number22
JournalSampling Theory, Signal Processing, and Data Analysis
Volume21
Number of pages7
ISSN2730-5716
DOIs
Publication statusPublished - 2023

Keywords

  • Approximate dual frames
  • Naimark’s Theorem
  • Operator representation of frames

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