On the structure of selfadjoint Toeplitz operators with rational matrix symbols

Abstract

Selfadjoint Toeplitz operators with rational matrix symbols are studied using a general result concerning functions ( T ( z ) − 1 x , y ) (T{(z)^{ - 1}}x,y) where T ( z ) T(z) is a polynomial family of Toeplitz operators with rational matrix symbols. It is proved that, apart from a finite number of points, these functions can be continued analytically across the boundary of the resolvent set of T ( z ) T(z) , for a dense set of x x ’s and y y ’s. This implies piecewise analyticity of the spectral measure ( E x , x ) (Ex,x) of selfadjoint Toeplitz operators with rational matrix symbol, for a dense set of x x ’s.