Frame properties of operator orbits. Issue 1 (24th October 2019) - Christensen, Ole; Hasannasab, Marzieh; Philipp, Friedrich
Abstract: We consider sequences in a Hilbert space H of the form ( T n f 0 ) n ∈ I, with a linear operator T, the index set being either I = N or I = Z, a vector f 0 ∈ 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 f 0 is ( T n f 0 ) 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 ( T n f 0 ) 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 ( T n f 0 ) n ∈ N with a bounded operator T is significantly larger than what could be expected from the examples known so far.