New York Journal of Mathematics
Volume 17a (2011) 39-44

  

M. Stessin, R. Yang, and K. Zhu

Analyticity of a joint spectrum and a multivariable analytic Fredhom theorem

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Published: January 30, 2011
Keywords: compact operators, Fredholm operators, analytic set, trace, determinant, joint spectrum
Subject: Primery 30H05, 30E10, Secondary 30D55

Abstract
For an n-tuple of compact operators T=(T1,...,Tn) on a Hilbert space H we consider a notion of joint spectrum of T, denoted by Σ(T), which consists of points z=(z1,...,zn) in Cn such that I+z1T1+...+znTn 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 Cn. 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