On unconditional bases in tensor products of Köthe echelon spaces (Q1371336)

It is well-known that the tensor product basis \(\{e_i\otimes e_j\}_{i,j\geq 1}\) of the tensor product \(\ell_p\otimes \ell_q\) for \(p= q=0\) or \(1< p\), \(q<\infty\), is not unconditional, where \(\{e_i\}\) is the canonical basis of \(\ell_p\). However, for Köthe sequences, the tensor product basis is unconditional in the projective tensor product \(\lambda_p(A)\otimes \lambda_q(B)\) of Köthe sequence spaces if one of the spaces is nuclear. Here, it is shown that if \(A= (a^n)\), \(B= (b^n)\) are Köthe matrices, then the tensor product basis of the tensor product \(\lambda_p(A)\otimes\lambda_q(B)\) is unconditional if and only if for each \(n\) there is an \(m\) such that for any bijection \(\sigma: \mathbb{N}\to\mathbb{N}\), the sequence \(\{a^n_i b^n_{\sigma(i)}/ a^m_i b^m_{\sigma(i)}\}\) is in \(\ell_1\).