Well-separating common complements of a sequence of subspaces of the same codimension in a Hilbert space are generic

TL;DR

This paper studies the existence and typicality of common complements for sequences of subspaces in Hilbert and Banach spaces, showing they are generally abundant under certain conditions related to exponential splittings.

Contribution

It establishes that common complements with subexponential decay are generic in Hilbert spaces and that the existence of one such complement in Banach spaces implies their genericity.

Findings

Common complements with subexponential decay are generic in Hilbert spaces.

Existence of one such complement in Banach spaces implies they are generic.

Results are motivated by applications to exponential type splittings like the multiplicative ergodic theorem.

Abstract

Given a family of subspaces we investigate existence, quantity and quality of common complements in Hilbert spaces and Banach spaces. In particular we are interested in complements for countable families of closed subspaces of finite codimension. Those families naturally appear in the context of exponential type splittings like the multiplicative ergodic theorem, which recently has been proved in various infinite-dimensional settings. In view of these splittings, we show that common complements with subexponential decay of quality are generic in Hilbert spaces. Moreover, we prove that the existence of one such complement in a Banach space already implies that they are generic.