The Numerical Range of a Weighted Shift

Quentin F. Stout
EECS Department, University of Michigan


Abstract: Let T be a weighted shift operator on a Hilbert space. We determine the numerical radius of T when the space is finite dimensional and T is a unilateral shift or a circular shift, and when the space is infinite dimensional and T is Hilbert-Schmidt, periodic, or a finite perturbation of periodic. For several of these cases we also determine when the numerical range is closed, completing the determination of the numerical range and answering a question of Ridge.

An important step is the determination of the eigenvalues of a self-adjoint tri-diagonal matrix with zeroes on its diagonal. A simple formula for the eigenvalues is given when the matrix is finite dimensional or Hilbert-Schmidt.

Keywords: weighted unilateral or bilateral shift, Hilbert space, numerical radius, numerical range, self-adjoint tri-diagonal matrix, circular shift, circularly symmetric functions

Complete paper. This paper appears in Proc. American Mathematical Society 88 (1983), pp. 495-502.

Additional work of mine in operator theory. However, I not longer do any research in this area.


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