05/06/2009, 15:00 — 16:00 — Room P3.10, Mathematics Building
Leiba Rodman, College of William and Mary, Williamsburg, VA, USA
Linear dependence of operators via sesquilinear forms
The numerical values (associated with the numerical ranges) and
\(q\)-numerical values (associated with the \(q\)-numerical ranges)
of two Hilbert space operators are compared. The main result of the
talk states that the absolute values of the \(q\)-numerical value
function of two operators coincide if and only if the operators are
unimodular scalar multiples of each other, for \(q\) positive and
less than one. In the extreme cases when \(q\) is equal to one or
to zero, additional possibilities occur. These statements are
framed in terms of \(C\)-numerical ranges where the operator \(C\)
is nonscalar of rank one. The results are motivated by an
application to the problem (still largely unsolved) of
characterizing norm preservers of Jordan products of matrices.
(Joint work with B. Kuzma, G. Lesnjak, C.-K. Li, T. Petek.)