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Journal of Operator Theory

Volume 42, Issue 2, Fall 1999  pp. 331-350.

Regular operators on Hilbert $C^*$-modules

Authors Arupkumar Pal
Author institution: Indian Statistical Institute, 7, SJSS Marg, 110 016 New Delhi, India

Summary:  A regular operator $T$ on a Hilbert $C^*$-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of $(I+T^*T)$ is dense. Semiregular operators are a slightly larger class of operators that may not have this property. It is shown that, like in the case of regular operators, one can, without any loss in generality, restrict oneself to semiregular operators on $C^*$-algebras. We then prove that for abelian $C^*$-algebras as well as for subalgebras of the algebra of compact operators, any closed semiregular operator is automatically regular. We also determine how a regular operator and its extensions (and restrictions) are related. Finally, using these results, we give a criterion for a semiregular operator on a liminal $C^*$-algebra to have a regular extension.

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