Additions to the Periodic Decomposition Theorem

Abstract

A pair of linear bounded commuting operators T₁, T₂ in a Banach space is said to possess a decomposition property (DePr) if

Ker (I-T₁)(I-T₂) = Ker (I-T₁) + Ker (I-T₂).

A Banach space X is said to possess a 2-decomposition property (2-DePr) if every pair of linear power bounded commuting operators in X possesses the DePr. It is known from papers of M. Laczkovich and Sz. Révész that every reflexive Banach space X has the 2-DePr.

In this paper we prove that every quasi-reflexive Banach space of order 1 has the 2-DePr but not all quasi-reflexive spaces of order 2. We prove that a Banach space has no 2-DePr if it contains a direct sum of two non-reflexive Banach spaces. Also we prove that if a bounded pointwise norm continuous operator group acts on X then every pair of operators belonging to it has a DePr.

A list of open problems is also included.