$$\ell ^p(G)$$ ℓ p ( G ) -Linear Independence and p-Zero Divisors

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.