Additions to the Periodic Decomposition Theorem
Source: Acta Mathematica Hungarica, Volume 90, Number 4, 2001 , pp. 293-305(13)
Abstract:A pair of linear bounded commuting operators T_1, T_2 in a Banach space is said to possess a decomposition property (DePr) if
Ker (I-T_1) (I-T_2) = Ker (I-T_1) + Ker (I-T_2).
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.
Document Type: Regular Paper
Publication date: January 1, 2001