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The numerical range of a weighted shift


Author: Quentin F. Stout
Journal: Proc. Amer. Math. Soc. 88 (1983), 495-502
MSC: Primary 47B37; Secondary 47A12
DOI: https://doi.org/10.1090/S0002-9939-1983-0699421-1
MathSciNet review: 699421
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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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0699421-1
Article copyright: © Copyright 1983 American Mathematical Society

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