Seminar on October 29 - Tirthankar Bhattacharyya

Spectral and Complete Spectral Sets

Prof. Tirthankar Bhattacharyya

Indian Institute of Science, Bangalore

Time: 12 to 1 pm, Tuesday, October 29, 2013
Venue: CR 201

Abstract: A compact set \(K\) in \({\Bbb C}^n\) is a spectral set for a commuting tuple \(T=(T_1,\dots,T_n)\) of bounded Hilbert space operators if the joint spectrum of \(T\) is contained in \(K\) and if the von Neumann-type inequality \[ ||r(T_1,\dots,T_n)|| \le \sup\{ |r(z)| : z\in K\} \] holds for all rational functions \(r\) in \(Rat(K)\). If the corresponding inequality holds even for every \(m\times m\) matrix \(r=(r_{ij})\) with entries in \(Rat(K)\) then \(K\) is called a complete spectral set for \(T\). By a result of Arveson, \(K\) is a complete spectral set for \(T\) if and only if \(T\) possesses a normal boundary dilation over \(K\). For nice domains in \(\Bbb C\), such as the unit disc or annulus, the conditions of being spectral and completely spectral are equivalent. In the lecture we start with the classical theory and try to present some more recent positive and negative results on one and several variable spectral sets.