ON RANK-ONE  ?  *-COMMUTING OPERATORS

Nuha Hamada

doi:10.24297/jam.v9i5.2345

ON RANK-ONE ? *-COMMUTING OPERATORS

Authors

Nuha Hamada Al Ain University of Science and Technology, Abu Dhabi campus

DOI:

https://doi.org/10.24297/jam.v9i5.2345

Keywords:

Hilbert space, rank one operator, Î» *-commutator, paranormal operator.

Abstract

Let ? be a non zero complex number. An operator A is a rank one ? *-commutes with B if AB - ? BA* has rank one. If, moreover, B is compact operator then A is called to belong to (H). In other words,(H) = fA 2 B(H)jAB-? BA* has rank one for some compact operator B} . We study the basic properties of (H). We prove that if A 2 B(H) has an eigenvalue dierent than? , and A has a fixed point then
(H).