| dc.description.abstract | A classical theorem of Issai Schur states that any n×n matrix is unitarily equivalent to
an upper-triangular matrix, and hence can be decomposed as the sum of a normal matrix
and a nilpotent matrix. Dykema, Sukochev and Zanin generalized this decomposition to
any operator in a von Neumann algebra with a normal, faithful, tracial state, replacing
nilpotent with s.o.t.-quasinilpotent.
In this paper we study the decomposition described by Dykema, Sukochev and Zanin.
We generalize the construction presented by Dykema, Sukochev and Zanin and introduce
the idea of a spectral ordering, a function Ø : [0; 1] → C which is suitable for construction
of such a decomposition. We give sufficient conditions for a function to be a spectral
ordering for an operator.
In the course of our investigation we develop the theory of SOT-quasinilpotent operators,
and construct an operator Q which is SOT-quasinilpotent and has a spectrum which
is a non-trivial interval of the real line; such an operator had not previously appeared in
the literature.
We then restrict ourselves to operators with finitely supported Brown measure and
investigate the properties of an operator T with quasinilotent upper-triangular part Q. We
show this is equivalent to several conditions, including decomposability (in the sense of
C. Foias) and having a finite spectrum. | en |
