Frame properties of operator orbits

Ole Christensen, Marzieh Hasannasab and Friedrich Philipp

Mathematische Nachrichten, 2020, vol. 293, issue 1, 52-66

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.

Date: 2020
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