 

M. Stessin, R. Yang, and K. Zhu
Analyticity of a joint spectrum and a multivariable analytic Fredhom theorem view print


Published: 
January 30, 2011

Keywords: 
compact operators, Fredholm operators, analytic set, trace, determinant, joint spectrum 
Subject: 
Primery 30H05, 30E10, Secondary 30D55 


Abstract
For an ntuple of compact operators T=(T_{1},...,T_{n}) on a Hilbert space H we consider a notion of joint spectrum of T, denoted by Σ(T), which consists of points z=(z_{1},...,z_{n}) in
C^{n} such that I+z_{1}T_{1}+...+z_{n}T_{n} is not invertible, where I is the identity operator on H. Using the theory of determinants for certain Fredholm operators we show that Σ(T) is always an analytic set of codimension 1 in
C^{n}. This result is in fact a special case of a multivariable version of the analytic Fredholm theorem.


Author information
M. Stessin:
Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12222
stessin@math.albany.edu
R. Yang:
Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12222
ryang@math.albany.edu
K. Zhu:
Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12222
kzhu@math.albany.edu

