Geodesic
The Collatz-Wielandt quotient for pairs of nonnegative operators
Applications of Mathematics, Tome 65 (2020) no. 5, pp. 557-597

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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.
DOI : 10.21136/AM.2020.0260-19
Classification : 15A22, 15A45, 15B48, 15B57, 94A40
Keywords: Perron-Frobenius theory; Collatz-Wielandt quotient; completely positive operator; commodity pricing; wireless network; quantum information theory 
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Friedland, Shmuel. The Collatz-Wielandt quotient for pairs of nonnegative operators. Applications of Mathematics, Tome 65 (2020) no. 5, pp. 557-597. doi: 10.21136/AM.2020.0260-19

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