Rescaling of unconditional Schauder frames in Hilbert spaces and completely bounded maps

Abstract

We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum _{k=1}^{\infty }⟨u,y_k⟩x_k,$$ then there exist nonzero scalars $\{{\alpha }_k{\}}_{k=1}^{\infty }$ such that both sequences $\{{\alpha }_k{x}_k{\}}_{k=1}^{\infty }$ and $\{{\overline{\alpha }}_k^{-1}{y}_k{\}}_{k=1}^{\infty }$ are frames. Our result has the following equivalent reformulation: if $\Phi :{\ell }^{\infty }\to B(H)$ is a bounded linear map such that for every element of the unit vector basis $e_k$ in ${\ell }^{\infty }$ the operator $\Phi (e_k)$ has rank one, then $\Phi$ is completely bounded.