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

TL;DR

This paper establishes conditions under which unconditional Schauder frames in Hilbert spaces can be rescaled to form frames, and connects these conditions to the complete boundedness of certain linear maps.

Contribution

It proves that unconditional Schauder frames can be rescaled into frames and characterizes the complete boundedness of specific bounded linear maps on Hilbert spaces.

Findings

Rescaling of unconditional Schauder frames into frames is possible under certain conditions.

A bounded linear map with rank-one images of basis vectors is necessarily completely bounded.

The results connect frame theory with operator space theory and completely bounded maps.

Abstract

We prove that if every element $u$ in a Hilbert space $H$ admits a representation as unconditionally convergent series $$u=\sum_{k=1}^\infty \langle u, y_k\rangle x_k,$$ 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.