An asymptotic variant of the Fuglede-Putnam theorem on commutators for elements of Banach algebras

Abstract

The Fuglede-Putnam theorem (in Moore's asymptotic form) on the commutators of normal operators of a Hilbert space is generalized, in particular, in the following form. Leta1,a1, b1 and b2 be the elements of a complex Banach algebra such that [a1 b1]=[a2, b2]=0 and\(\left\| {e^{\bar \lambda a_2 - \lambda b_2 } } \right\| = o\left( {\left| \lambda \right|^{1'_2 } } \right)\) as λ → ∞. Then the inequality ∥b1x−xb2∥⩽ϕ(∥a1x−xa2∥), where ϕ (e) → υ as e → 0, holds uniformly in every ball ∥x∥⩽R<∝.