Journal of Operator Theory

Volume 48, Issue 2, Fall 2002 pp. 235-253.

K-groups of Banach algebras and strongly irreducible decompositions of operators

Authors: Yang Cao (1), Junsheng Fang (2), and Chunlan Jiang (2)
Author institution: (1) Department of Mathematics, Jilin University, Chang chun 130023, China
(2) Academia Sinica, Institute of Mathematics, Beijing 100080, China
(3) Department of Applied Mathematics, Hebei University of Technology, Tianjin 300130, China

Summary: A bounded linear operator $T$ on the Hilbert space $\sH$ is called strongly irreducible if $T$ does not commute with any nontrivial idempotent operator. One says that $T$ has a finite $\SI$ decomposition if $T$ can be written as the direct sum of finitely many strongly irreducible operators. In this paper, we use the ${\rm K}_0$-group of the commutant of operators to characterize operators with unique finite $\SI$ decomposition up to similarity. Also we show that the ${\rm K}_0$-group of $H^\infty(\Omega)$ is isomorphic to the integers, where $\Omega$ is simply connected.

Keywords: ${\rm K}_0$-group, ${\rm (SI)}$ decomposition, commutant of operators

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