*Research Paper*

Year: 2017 | Month: June | Volume: 4 | Issue: 6 | Pages: 31-43

**The Strongly Compact Algebras**

Osman Abdalla A. OS^{1},Yagoub Hussein Ahmed. H^{2}

^{1}Department of Mathematics - College of Science & Arts in OklatAlskoor - Qassim University - Saudi Arabia

^{2}Department of Mathematics- College of Science -University of Sudan

Corresponding Author: Osman Abdalla A. OS

**ABSTRACT**

Algebra of bounded linear operators on a Hilbert space is strongly compact if its unit ball is relatively compact in the strong operator topology. The algebra generated by the operator and the identity is strongly compact. This notion was introduced by Lomonosov as an approach to the invariant subspace problem for essentially normal operators. The basic properties of strongly compact algebras are established. A characterization of strongly compact normal operators is provided in terms of their spectral representation, and some applications. Finally, necessary and sufficient conditions for a weighted shift to be strongly compact are obtained in terms of the sliding products of its weights, and further applications are derived.

*Key words: ** strongly compact, Hilbert space, operators, subalgebra, relatively, commutant, topology.*

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