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

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## 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 |
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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 |

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