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A collection & of operators on a finite-dimensional space is said to be simultaneously triangularizable if there is a basis of the space with respect to which the matrices of all members of ¥ are upper triangular. Equivalently, there exists a maximal totally ordered set of subspaces which are invariant under all members of Sf. More generally, a collection ¥ of bounded operators on a Banach space X is called simultaneously triangularizable if there is a maximal totally ordered complete set of (closed) subspaces of X(a maximal nest) which are ^-invariant. Several authors have recently worked on the problem of simultaneous triangularizability. In this paper, we obtain some simple necessary and sufficient conditions for simultaneous triangularizability of a (not necessarily norm-closed) algebra of compact operators on a Banach space. This improves on results of [5,7], and allows us to apply a theorem of Aupetit [1] to the present context. We also extend to Banach-space operators a result of [10] on simultaneous triangularizability of semigroups of traceclass operators on a Hilbert space. Using our first theorem, we are able to present a proof which is considerably shorter, even in the Hilbert-space case treated in [10]. We include a result on triangularizability of certain algebras of Hilbert-space operators that 'approximate' maximal abelian von Neumann algebras.

We introduce the notion of C -orbit reflexivity and study its properties. An operator on a finite-dimensional space is C -orbit reflexive if and only if the two largest blocks in its Jordan form corresponding to nonzero eigenvalues with the largest modulus differ in size by at most one. Most of the proofs of our results in infinite dimensions are obtained from purely algebraic results we obtain from linear-algebraic analogs of C -orbit reflexivity. Mathematics subject classification (2010): Primary 05C38, 15A15; Secondary 05A15, 15A18.