Triangularizability of polynomially compact operators (Q814349)

A (linear) operator \(T\) on a complex Banach space is said to be polynomially compact if the operator \(p(T)\) is compact for any nonzero polynomial \(p\). If \(T^k\) is compact for some \(k \in {\mathbb{N}}\), then \(T\) is said to be power compact. The monic polynomial \(p\) of the smallest degree for which \(p(T)\) is compact is called the minimal polynomial of \(T\). In Section 1 of the paper under review, the author reveals that polynomially compact operators inherit many of the well-known spectral properties of compact operators involving the minimal polynomial. An example shows that the set of polynomially compact operators, in general, is not closed under either addition or multiplication. However, by introducing the concept of essentially commuting sets of operators (i.e., operators with compact commutators), it is shown that for finite sets of essentially commuting polynomially compact operators, the polynomial compactness of a non-commutative polynomial of such a set is ensured. In Section 2, algebras of polynomially compact operators are investigated, including the concepts of (simultaneous) triangularisability and reducibility. In Section 3, the author asks for sufficient conditions which imply reducibility and triangularisability for semigroups of essentially commuting power compact operators. In the final Section 4 of the paper, the structure of polynomially compact operators on Hilbert spaces is illuminated, thus extending the structure theorem of \textit{C. L. Olsen} [Am.\ J.\ Math.\ 93, 686--698 (1971; Zbl 0239.47017)] to the non-separable case.