In this paper, we study operators for which T*T and T + T* commute. Let Θ denote the class of such operators in L(H). We show that every operator in Θ is subscalar of order 4. From this result, we give partial solutions to the invariant subspace problem. In addition, we examine some extensions of operators in Θ. Finally, we prove that if T belongs to Θ, then Weyl's theorem holds for T and the spectral mapping theorem corresponding to the Weyl spectrum is satisfied for T.
|Number of pages||12|
|Journal||Bulletin of the London Mathematical Society|
|State||Published - Aug 2013|