International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 42, Pages 2667-2672
doi:10.1155/S0161171203012043
Local spectral theory for 2×2 operator matrices
Département de Mathématiques et Informatique, Faculté des Sciences de Rabat, Université Mohamed V, Rabat BP 1014, Morocco
Received 13 February 2001; Revised 11 July 2001
Copyright © 2003 H. Elbjaoui and E. H. Zerouali. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We discuss the spectral properties of the operator MC∈ℒ(X⊕Y) defined by MC:=(AC0B), where A∈ℒ(X), B∈ℒ(Y), C∈ℒ(Y,X), and X, Y are complex Banach spaces. We prove that (SA∗∩SB)∪σ(MC)=σ(A)∪σ(B) for all C∈ℒ(Y,X). This allows us to give a partial positive answer to Question 3 of Du and Jin (1994) and generalizations of some results of Houimdi and Zguitti (2000). Some applications to the similarity problem are also given.