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∗-modulesAuthors: 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 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 A any densely defined A-linear closed operator t between Hilbert C∗-modules has polar decomposition, if and only if any densely defined A-linear closed operator t between Hilbert C∗-modules has generalized inverse, if and only if A is a C∗-algebra of compact operators.
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