Frame properties of operator orbits

By Ole W. Christensen, Marzieh Hasannasab, Friedrich M. Philipp. Mathematische Nachrichten, 2018

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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.

Subfields: Applied Mathematics, Mathematical Physics

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