Multipliers on generalized mixed norm sequence spaces (Q1725452)

Summary: Given \(1 \leq p\), \(q \leq \infty\) and sequences of integers \((n_k)_k\) and \((n_k')_k\) such that \(n_k \leq n_k' \leq n_{k + 1}\), the generalized mixed norm space \(\ell^{\mathcal{I}}(p, q)\) is defined as those sequences \((a_j)_j\) such that \(((\sum_{j \in I_k} |a_j|^p)^{1 / p})_k \in \ell^q\) where \(I_k = \{j \in \mathbb N_0 \text{s.t.} n_k \leq j < n_k' \}\), \(k \in \mathbb N_0\). The necessary and sufficient conditions for a sequence \(\lambda = (\lambda_j)_j\) to belong to the space of multipliers \((\ell^{\mathcal{I}}(r, s), \ell^{\mathcal{J}}(u, v))\), for different sequences \(\mathcal{I}\) and \(\mathcal{J}\) of intervals in \(\mathbb N_0\), are determined.