Abstract: Let T be a weighted shift operator on a Hilbert space. We compute the numerical radius of T when T is finite, circular, 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. We give a simple formula for the eigenvalues when the matrix is finite dimensional or Hilbert-Schmidt.

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

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

Additional work in operator theory

Copyright © 2006-2008 Quentin F. Stout