Journal of Operator Theory

Volume 63, Issue 2, Spring 2010 pp. 271-282.

Adjointability of densely defined closed operators and the Magajna--Schweizer theorem

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 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 $P_FP_{G(t)^\bot}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^*$-algebra $\mathcal A$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules is regular, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules admits a densely defined adjoint operator, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained.

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