Localizations of the Kleinecke-Shirokov theorem

Abstract

Abstract. A local version of the Kleinecke-Shirokov theorem is proved. The results easily extend to bounded linear derivations on Banach algebras. In case of algebraic elements, an improved bound on the nilindex of a commutator is obtained as a consequence.Mathematics subject classification (2000): 47B47, 47B48, 46H05.

Triangularizability of families of polynomially compact operators

“…We may assume that F contains a nonscalar operator. If F [1] = {0}, then F is reducible, by Corollary 2.3. So, we may assume that…”

“…Suppose now that F consists of algebraic operators. In this case the commutator [A, B] is nilpotent, by Jacobson's lemma (see, e.g., [1]). Since every operator in F commutes with [A, B], the kernel ker([A, B]) is a nontrivial closed subspace that is invariant under F .…”

“…In particular, if F [1] = {0}, then operators from F commute. Note that every nilpotent Lie algebra is L-nilpotent.…”

“…Proof. Consider first the case when F [1] = {0}. Let us denote by d(T ) the degree of the minimal polynomial of an algebraic operator T .…”

The Matrix Equation XA- AX=Xαg(X) over Fields or Rings

“…Let A ∈ M n (R) be similar over R to diag(λ 1 I n1 , · · · , λ r I nr ) with n 1 + · · · + n r = n and such that, for every i = j, λ i − λ j is not a zero-divisor. Let X be a nilpotent solution of Eq (8). i) Then there is P ∈ GL n (R) such that A = P diag(λ 1 I n1 , · · · , λ r I nr ) P −1 and X = P diag(X 1 , · · · , X r ) P −1…”