\((E,F)\)-Schur multipliers and applications (Q2840502)

Let \(c_0\) be the linear space of all real sequences converging to zero. For \(x=(x_i)_{i=1}^{\infty}\in c_0\), let \(|x|=(|x_i|)_{i=1}^{\infty}\) and \(x^*=(x_{i}^{*})_{i=1}^{\infty}\) be the non-increasing rearrangement of \(|x|\), that is, \(x_{i}^{*}=|x_{n_i}|\), where \((n_i)_{i=1}^{\infty}\) is a permutation of positive integers such that the sequence \((|x_{n_i}|)_{i=1}^{\infty}\) is non-increasing. A~linear subspace \(E\subseteq c_0\) equipped with a Banach norm \(\| \cdot \|\) is a symmetric sequence space if it satisfies the following conditions: (i) if \(x, y\in E\) and \(|x|\leq |y|\), then \(\| x\| \leq \| y\|\); (ii) if \(x\in E\), then \(x^*\in E\) and \(\| x^*\|=\| x\|\). It is clear that \(\ell_p\) (\(1\leq p< \infty\)) are symmetric sequence spaces. For a symmetric sequence space \(E\), its Köthe dual is NEWLINE\[NEWLINEE^\times=\{ y\in \ell_\infty: \sum_{i=1}^{\infty}|x_iy_i|<\infty \text{ for all } x\in E\}.NEWLINE\]NEWLINE If \(E^\times\) equals the topological dual \(E^*\) of \(E\), then each bounded linear operator \(A\) from \(E\) to a symmetric sequence space \(F\) can be identified with a matrix \([a_{ij}]_{i,j=1}^{\infty}\) (each row represents an element from \(E^\times\) and each column represents an element from \(F\)). An infinite matrix \(M=[m_{ij}]_{i,j=1}^{\infty}\) is called an \((E,F)\)-Schur multiplier if, for every \(A\in B(E,F)\) which is represented by a matrix \([a_{ij}]_{i,j=1}^{\infty}\), the Schur product \(M\ast A=[m_{ij}a_{ij}]_{i,j=1}^{\infty}\) is in \(B(E,F)\) as well.NEWLINENEWLINEThe paper under review generalizes some results about Schur multipliers proved by \textit{G. Bennett} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 34, 135--155 (1976; Zbl 0351.40002); Duke Math. J. 44, 603--639 (1977; Zbl 0389.47015)] for the classical sequence spaces. Several results are obtained asserting continuous embedding of the \((E,F)\)-multiplier space into the classical \((p,q)\)-multiplier space. Furthermore, many examples of symmetric sequence spaces \(E\) and \(F\) are presented whose projective and injective tensor products are not isomorphic to any subspace of a Banach space with an unconditional basis.