BOOK-CHAPTER

Singly generated algebras containing a compact operator

Abstract

For an operator T, consider P (T), the norm closed algebra generated by T and the identity operator. The question is, "When does this algebra contain a compact operator?". The question remains unanswered. However it is shown that the set of all operators such that P (T) contains a compact operator is dense in \(\mathcal{B}(\mathcal{H})\) and the interior of this set is characterized. If the term "compact operator" is replaced by "finite rank operator" or if P (T) is replaced by the weakly closed or weak-star closed algebra generated by T and the identity, the same results are obtained. Similar questions are raised and answered for other algebras associated with an operator.

Keywords:
Operator (biology) Operator algebra Compact operator Mathematics Finite-rank operator Pure mathematics Algebra over a field Shift operator Ladder operator Rank (graph theory) Discrete mathematics Combinatorics Computer science Chemistry Banach space

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Topics

Holomorphic and Operator Theory
Physical Sciences →  Mathematics →  Applied Mathematics
Advanced Banach Space Theory
Physical Sciences →  Mathematics →  Mathematical Physics
Advanced Operator Algebra Research
Physical Sciences →  Mathematics →  Mathematical Physics

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