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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