Commutators and normal operators

Crabb, M. J. and Spain, P. G. (1977) Commutators and normal operators. Glasgow Mathematical Journal, 18(2), pp. 197-198. (doi: 10.1017/S001708950000327X)

Abstract

Let X be a Banach space and L (X) the Banach algebra of bounded linear operators on X. An operator Tin L (X) is hermitian if || e itT \\ = 1 (teR), and is normal ifT=R+//where R and / are commuting normal operators; R and / are then determined uniquely by T, and we may write T* = R—iJ. These definitions extend those for operators on Hilbert spaces. More details may be found in [1]. Given T in L(X) we may define the left-multiplication operator kT: L{X)-*L(X): Av^TA and the right-multiplication operator pT : L{X) -> L(X) : A-* AT. It is easy to check (see [2], for instance) that kT and pT are hermitian in L(L{X)) if 7"is hermitian in L(X). It follows that Ajy, — pNl is normal in L(L(X)) if Nt and N2 are normal in L(X). Putnam [4] proved that if H is a Hilbert space, if A,BeL(H), and if A is normal and commutes with AB—BA, then A commutes with B. The following result extends Putnam's theorem to operators on Banach spaces.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Crabb, Dr Michael and Spain, Dr Philip
• Authors: Crabb, M. J., and Spain, P. G.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Glasgow Mathematical Journal
• Publisher: Cambridge University Press
• ISSN: 0017-0895
• ISSN (Online): 1469-509X