John B. ConwayGabriel T. Prǎjiturǎ
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.
John B. ConwayWacław Szymański
Hari BercoviciCiprian FoiaÅCarl Pearcy