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

Abstract

Given a finite orthonormal sequence $\Phi_n = (φ₁,·s,φ_n)$ in some $L₂(\mu)$ and vectors $x1,·s, x_n$ in some Banach space X we are interested in the norm of the sums $∑_{j=1}ⁿ φ_j(t)x_j$ in $L₂^X(\mu)$.A constuction in [1] suggests that the system $\phi_n$ may be replaced by the set $∏_n=(?₁,·s?_n)$ of coordinate functions $?_j(σ₁, ·s, σ_n)=σ_j$ on $?^n-1$ viewed as un orthonormal system with respect to a suitable measure λ on $?^n-1$.We show by a convolutional argument that after symmetrization the measure λ is uniquely determined.We also discuss related questions.