Frame properties of operator orbits. Issue 1 (24th October 2019)

• Record Type: Journal Article
• Authors: Christensen, Ole; Hasannasab, Marzieh; Philipp, Friedrich
• Is Part Of: Mathematische Nachrichten. Volume 293:Issue 1(2020)
• Page Start: 52
• Page End: 66
• Publication Date: 2019-10-24
• Subjects: contraction -- frame -- Gabor frame -- operator orbit -- 94A20 -- 42C15 -- 30J05; Mathematics -- Periodicals; 510.5
• DOI: 10.1002/mana.201800344
• Languages: English
• ISSNs: 0025-584X

Abstract

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.