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Frame properties of operator orbits

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  • Ole Christensen
  • Marzieh Hasannasab
  • Friedrich Philipp

Abstract

We consider sequences in a Hilbert space H of the form (Tnf0)n∈I, with a linear operator T, the index set being either I=N or I=Z, a vector f0∈H, and answer the following two related questions: (a) Which frames for H are of this form with an at least closable operator T? and (b) For which bounded operators T and vectors f0 is (Tnf0)n∈I a frame for H? As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form (Tnf0)n∈N with a bounded operator T. The corresponding problem for I=Z remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form (Tnf0)n∈N with a bounded operator T is significantly larger than what could be expected from the examples known so far.

Suggested Citation

  • Ole Christensen & Marzieh Hasannasab & Friedrich Philipp, 2020. "Frame properties of operator orbits," Mathematische Nachrichten, Wiley Blackwell, vol. 293(1), pages 52-66, January.
    • Handle: RePEc:bla:mathna:v:293:y:2020:i:1:p:52-66
    DOI: 10.1002/mana.201800344
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