Geometric, spectral and asymptotic properties of averaged products of projections in Banach spaces

TL;DR

This paper extends classical convergence results of projection iterates from Hilbert spaces to complex Banach spaces, using spectral criteria and geometric properties like uniform convexity or smoothness.

Contribution

It generalizes the convergence theorems for products and convex combinations of projections from Hilbert spaces to Banach spaces with specific geometric conditions.

Findings

Strong convergence of projection iterates in Banach spaces under geometric assumptions

Extension of von Neumann-Halperin and Lapidus theorems beyond Hilbert spaces

Use of boundary spectrum criteria for convergence proofs

Abstract

According to the von Neumann-Halperin and Lapidus theorems, in a Hilbert space the iterates of products or, respectively, of convex combinations of orthoprojections are strongly convergent. We extend these results to the iterates of convex combinations of products of some projections in a complex Banach space. The latter is assumed uniformly convex or uniformly smooth for the orthoprojections, or reflexive for more special projections, in particular, for the hermitian ones. In all cases the proof of convergence is based on a known criterion in terms of the boundary spectrum.