Frame properties of operator orbits

We consider sequences in a Hilbert space H of the form (Tnf0)n∈I is an element of I, with a linear operator T, the index set being either I=N or I=Z, a vector f0 is an element of 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 is an element of 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 is an element of 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∈I is an element of N with a bounded operator T is significantly larger than what could be expected from the examples known so far.