Acta Mathematica Hungarica

, Volume 90, Issue 4, pp 293–305

Additions to the Periodic Decomposition Theorem

  • V. M. Kadets
  • B. M. Shumyatskiy
Article

DOI: 10.1023/A:1010631129811

Cite this article as:
Kadets, V.M. & Shumyatskiy, B.M. Acta Mathematica Hungarica (2001) 90: 293. doi:10.1023/A:1010631129811

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.

Copyright information

© Kluwer Academic Publishers/Akadémiai Kiadó 2001

Authors and Affiliations

  • V. M. Kadets
    • 1
  • B. M. Shumyatskiy
    • 2
  1. 1.KHARKOV STATE UNIVERSITYKHARKOV, SVOBODY SQ., 4UKRAINE
  2. 2."MODEL" COMPANYKHARKOV, KOSMICHESKAYA UL., 26UKRAINE