Additions to the Periodic Decomposition Theorem

Authors: Kadets, V.M.1; Shumyatskiy, B.M.2

Source: Acta Mathematica Hungarica, Volume 90, Number 4, 2001 , pp. 293-305(13)

Publisher: Springer

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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

Affiliations: 1: KHARKOV STATE UNIVERSITY,310077 KHARKOV,SVOBODY SQ., 4,UKRAINE 2: "MODEL" COMPANY,310145 KHARKOV,KOSMICHESKAYA UL., 26,UKRAINE

Publication date: January 1, 2001

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