Joint spectral radius, operator semigroups, and a problem of W. Wojtyński
Recently, the second author proved a remarkable theorem: Every semigroup of compact quasinilpotent (i.e. Volterra) Banach space operators has a nontrivial invariant subspace [J. Funct. Anal. 162, No. 2, 313-322 (1999; Zbl 0921.47008)]. The essence of the proof is that the algebra generated by the semigroup also consists of quasinilpotent operators. Algebras of compact quasinilpotent operators have invariant subspaces, according to V. I. Lomonosov's theory. The main tool used by Turovskii was the joint spectral radius technique developed by him. The present paper continues the research on the connection between invariant subspace theory and the joint spectral radius technique. \textit{W. Wojtyński} [Stud. Math. 49, 263-273 (1977; Zbl 0346.46050)] showed that locally finite Lie algebras of compact operators have invariant subspaces. The question remained open for general Lie algebras of Volterra operators. This question is answered in the affirmative here. The authors use semigroups of principle operators (operators whose spectral radius coincides with their essential spectral radius). It is shown that if one such semigroup contains the group \(\{e^{tT}, t\in\mathbb{R}\}\) for some nonzero Volterra operator \(T\), then it has a hyperinvariant subspace. This gives the existence of hyperinvariant subspaces for Lie algebras of Volterra operators. The paper contains many accompanying results which are of independent interest.
locally finite Lie algebras of compact operators
scientific article; zbMATH DE number 1556913
scientific article