Journal of Operator Theory

Volume 64, Issue 2, Fall 2010 pp. 377-386.

Generalized inverses and polar decomposition of unbounded regular operators on Hilbert $C^*$-modules

Authors: Michael Frank (1) and Kamran Sharifi (2)
Author institution:(1) Hochschule fuer Technik, Wirtschaft und Kultur (HTWK) Leipzig, Fachbereich IMN, Gustav-Freytag-Strasse 42A, D-04277 Leipzig, Germany
(2) Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161-316, Shahrood, Iran

Summary: In this note we show that an unbounded regular operator $t$ on Hilbert $C^*$-modules over an arbitrary $C^*$-algebra $ \mathcal{A}$ has polar decomposition if and only if the closures of the ranges of $t$ and $|t|$ are orthogonally complemented, if and only if the operators $t$ and $t^*$ have unbounded regular generalized inverses. For a given $C^*$-algebra $ \mathcal{A}$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has polar decomposition, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules has generalized inverse, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators.

Keywords: Hilbert $C^*$-module, unbounded operator, polar decomposition, generalized inverses, $C^*$-algebras of compact operator

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