Simultaneous triangularization
Simultaneous triangularization of operators can be regarded as a kind of generalized commutativity: it implies (and is implied by) certain consequences of commutativity. It a priori requires the existence of common invariant subspaces [concerning this problem, the authors' book ``Invariant subspaces'' (1973; Zbl 0269.47003) became famous], and thus it particularly applies to operators that are closely related to compact operators. Of course, the concept originates from finite dimension. The first five chapters of the book are devoted to this classical development, with much modern insight. The sixth chapter presents basic facts on compact operators (including Hilden's proof of Lomonosov's theorem), and the next two chapters show that most of the finite dimensional results have satisfactory extensions to compact operators. On the other hand, the last ninth chapter provides a number of counterexamples in the case of general infinite-dimensional operators. The authors have done a great job in finding a simple presentation of a beautiful theory, and collecting all relevant references. Their own results and simplifications represent a substantial part of the book. Each chapter ends with corresponding historical remarks. One can find even the most recent results, in particular, Turovskiĭ's theorem: A semigroup of compact quasi-nilpotent operators on a Banach space is triangularizable. Nevertheless, this interesting development does not finish with the book. For instance, the authors have recently found even a simpler proof of Theorem 2.3.10 in their preprint ``On commutators of idempotents'' (2000). Other papers related to the book have appeared, for instance: \textit{R. Drnovšek} [Hyperinvariant subspaces for operator semigroups with commutators of rank at most one, Houston J. Math. 26, No. 3, 543-548 (2000; Zbl 0981.47005)]; \textit{R. Yahagi} [On simultaneous triangularization of commutants, Acta Sci. Math. 66, No. 3-4, 711-718 (2000; Zbl 1062.47504)]; \textit{G. MacDonald} [Constructing irreducible semigroups of idempotents, Acta Sci. Math. 67, 315-336 (2001)]. As a kind of generalized commutativity, the simultaneous triangularizability is closely related with nice behaviour of the spectrum and the spectral radius like submultiplicativity or sublinearity, trace properties, quasi-nilpotent commutators etc. These properties were studied earlier for Banach algebras, and it is interesting to see in the present book how the underlying algebraic structure (algebra or semigroup) influences their mutual relations. It seems that for simultaneous triangularizability the semigroup structure of the given family is a natural one. In addition to that, various additional properties of the members of the family are considered (idempotents, nilpotents, properties of commutators etc.). Altogether the book contributes significantly towards understanding these basic concepts.
scientific article; zbMATH DE number 1404753