Abstract
Let T be a dual integrable representation of a countable discrete LCA group G, acting on a Hilbert space \({\mathcal {H}}\). We consider the problem of characterizing \(\ell ^p(G)\)-linear independence (\(p\ne 2\)) of the system \(\{T_{k}\psi :k\in G\}\) for the given \(\psi \in {\mathcal {H}}\), which we previously studied in the context of the integer translates of a square integrable function. The extensions of the known results for translates to this setting are obtained using a slightly different approach, through which we show that, under certain conditions, this problem is related to the ‘Wiener’s closure of translates’ problem and the problem of the existence of p-zero divisors, arising around the zero divisor conjecture in algebra. Using this connection, we also obtain several improvements for the case of the integer translates.
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Slamić, I. \(\ell ^p(G)\)-Linear Independence and p-Zero Divisors. Mediterr. J. Math. 15, 120 (2018). https://doi.org/10.1007/s00009-018-1167-z
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DOI: https://doi.org/10.1007/s00009-018-1167-z
Keywords
- Bracket function
- cyclic vector
- dual integrable representation
- \(\ell ^p(G)\)-linear independence
- zero divisor