Spectral properties of elements of normed algebras and invariant subspaces
Let A be a normed algebra with norm \(\| \|_ A\), \(M\subset A\) be a bounded subset, M' be the commutant of M, A(M) be the subalgebra of A, generated by M, \(M^ n\) be the set of all products of n elements from M, \(\| M\| =\sup \{\| a\|_ A:a\in M\}\) and \(r(M)=\lim \| M^ n\|^{1/n}.\) An element a from a subalgebra B of A is called Lie radical if \(r(ab-ba)=0\) for each \(b\in B.\) In addition to the other results the conditions in order that the union of two triangulable families of compact operators from L(X) (the algebra of all bounded linear operators on a Banach space X with dim X 2) is triangulable and for given subsets S,M\(\subset L(X)\) the algebra A(S\(\cup M)\) and its commutant have a common nontrivial invariant subspace are described. Moreover it is stated that any Lie radical compact operator from a subalgebra \(B\subset L(X)\) is Lie radical for \(cl_{L(X)}A(B\cup B').\)
triangulable families of compact operators
scientific article; zbMATH DE number 3909129