Applications of Mathematics, Vol. 65, No. 5, pp. 557-597, 2020


The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland

Received October 10, 2019.   Published online June 30, 2020.

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.
Keywords:  Perron-Frobenius theory; Collatz-Wielandt quotient; completely positive operator; commodity pricing; wireless network; quantum information theory
Classification MSC:  15A22, 15A45, 15B48, 15B57, 94A40
DOI:  10.21136/AM.2020.0260-19

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Affiliations:   Shmuel Friedland, Department of Mathematics, Statistics and Computer Science, University of Illinois, 851 S. Morgan Street, Chicago, IL, 60607-7045, U.S.A., e-mail: friedlan@uic.edu


 
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