When may two systems of orthonormal functions be interchanged in vector-valued orthogonal sums?

Abstract

Given a finite orthonormal sequence(Error rendering LaTeX formula) in some and vectors in some Banach space X we are interested in the norm of the sums(Error rendering LaTeX formula) in .A constuction in [1] suggests that the system may be replaced by the set(Error rendering LaTeX formula) of coordinate functions(Error rendering LaTeX formula) on viewed as un orthonormal system with respect to a suitable measure λ on .We show by a convolutional argument that after symmetrization the measure λ is uniquely determined.We also discuss related questions.