The Spectrum and Essential Spectrum of Toeplitz Operators

Series: 
Analysis Seminar
Wednesday, March 6, 2013 - 14:00
1 hour (actually 50 minutes)
Location: 
Skiles 005
,  
Vanderbilt University
Organizer: 
On the Hardy space, by means of an elegant and ingenious argument, Widom showed that the spectrum of a bounded Toeplitz operator is always connected and Douglas showed that the essential spectrum of a bounded Toeplitz operator is also connected. On the Bergman space, in 1979, G. McDonald and the C. Sundberg showed that the essential spectrum of a Toeplitz operator with bounded harmonic symbol is connected if the symbol is either real or piecewise continuous on the boundary. They asked whether the essential spectrum of a Toeplitz operator on the Bergman space with bounded harmonic symbol is connected.  In this talk,  we will show an example that the spectrum and the essential spectrum of a Toeplitz operator with bounded harmonic symbol is disconnected. This is a joint work with Carl Sundberg.