Joint spectral radius and invariant subspaces (Q1580177)

The paper is a review of some recent results of the authors (no proofs are given) on existence of nontrivial invariant and hyperinvariant subspaces for some families of Banach space operators. For a bounded set \(M\) of elements of a normed algebra \(A\) put \(\|M\|=\{\|a\|\:a\in M\}\). Let \(M^n\) be the set of all products of \(n\) elements of \(M\). \(\rho(M)=\inf\|M^n\|^{1/n}\) is called the spectral radius of \(M\). The essential spectral radius of \(M\) is the spectral radius of the image of \(M\) in the Calkin algebra. Most of the results of the paper contain conditions on \(\rho(m)\) and/or \(\rho_e(M)\). The authors gave answers to questions 4 and 5 posed by \textit{W. Wojtynski} [Stud. Math. 59, 263-273 (1977; Zbl 0346.46050.)].