The numerical range of a weighted shift

Stout, Quentin F.

doi:10.1090/S0002-9939-1983-0699421-1

The numerical range of a weighted shift
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by Quentin F. Stout PDF

Proc. Amer. Math. Soc. 88 (1983), 495-502 Request permission

Abstract:

Let $T$ be a weighted shift 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 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.

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