Operator-valued Fourier–Haar multipliers on vector-valued $${L^1}$$ spaces II: a characterisation of finite dimensionality

Abstract

A necessary and sufficient condition is given to ensure the boundedness of Fourier–Haar multiplier operators from $$L^1 ([0, 1], X)$$ to $$L^1 ([0, 1], Y)$$ , where X is an arbitrary finite dimensional Banach space and Y is an arbitrary Banach space. The Fourier–Haar multiplier sequences come not from $${\mathbb {R}}$$ , as in the classical case, but from the space of bounded operators from the Banach space X to the Banach space Y. Moreover, it is shown that this condition characterises the finite dimensionality of the Banach space X.