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The Collatz-Wielandt Quotient for Pairs of Nonnegative Operators

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Abstract

In this paper we consider two versions of the Collatz-Wielandt quotient for a pair of nonnegative operators A, B that map a given pointed generating cone in the first space into a given pointed generating cone in the second space. If the two spaces and two cones are identical, and B is the identity operator, then one version of this quotient is the spectral radius of A. In some applications, as commodity pricing, power control in wireless networks and quantum information theory, one needs to deal with the Collatz-Wielandt quotient for two nonnegative operators. In this paper we treat the two important cases: a pair of rectangular nonnegative matrices and a pair of completely positive operators. We give a characterization of minimal optimal solutions and polynomially computable bounds on the Collatz-Wielandt quotient.

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The author thanks the referee for his comments.

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Computer Science, University of Illinois, 851 S. Morgan Street, Chicago, IL, 60607-7045, USA

    Shmuel Friedland

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  1. Shmuel Friedland
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Correspondence to Shmuel Friedland.

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S. Friedland was partially supported by Simons collaboration grant for mathematicians.

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Friedland, S. The Collatz-Wielandt Quotient for Pairs of Nonnegative Operators. Appl Math 65, 557–597 (2020). https://doi.org/10.21136/AM.2020.0260-19

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