Seminário Análise Funcional e Aplicações 

05/06/2009, 15:00 — 16:00 — Sala P3.10, Pavilhão de Matemática
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.)