The numerical range of a weighted shift

Abstract

Let T T be a weighted shift on a Hilbert space. We compute the numerical radius of T T when T T is finite, circular, Hilbert-Schmidt, periodic, or a finite perturbation of periodic. For several cases we also determine whether 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 selfadjoint tri-diagonal matrix with zeroes on its diagonal. We give a simple formula for the eigenvalues when the matrix is finite dimensional or Hilbert-Schmidt.