Operator-valued Fourier-Haar multipliers on vector-valued \({L^1}\) spaces. II: A characterisation of finite dimensionality
The paper under review is concerned with the boundedness of operator-valued Fourier-Haar multipliers from \(L^1([0,1],X)\) to \(L^1([0,1],Y)\), where \(X\) and \(Y\) are Banach spaces. A necessary and sufficient condition for the boundedness is obtained when \(X\) is finite dimensional. It is also shown that the finite dimensionality of \(X\) is necessary for such a characterization to hold. Moreover, a lower bound for the operator norm is obtained for any Banach space \(X\) whose dual has cotype \(q\) for some \(2\leq q<\infty\). The results complement \(L^p([0,1],X)\) to \(L^q([0,1],Y)\) boundedness results obtained by \textit{M. Girardi} [J. Math. Anal. Appl. 325, No. 2, 1314--1326 (2007; Zbl 1105.42024)].
scientific article; zbMATH DE number 7382081
scientific article