Journal of Operator Theory
Volume 63, Issue 2, Spring 2010 pp. 271-282.
Adjointability of densely defined closed operators and the Magajna--Schweizer theoremAuthors: 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 unbounded regular operators on Hilbert C∗-modu\-les over arbitrary C∗-algebras are discussed. A densely defined operator t possesses an adjoint operator if the graph of t is an orthogonal summand. Moreover, for a densely defined operator t the graph of t is orthogonally complemented and the range of PFPG(t)⊥ is dense in its biorthogonal complement if and only if t is regular. For a given C∗-algebra A any densely defined A-linear closed operator t between Hilbert C∗-modules is regular, if and only if any densely defined A-linear closed operator t between Hilbert C∗-modules admits a densely defined adjoint operator, if and only if A is a C∗-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained.
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