Applications of Mathematics, Vol. 65, No. 5, pp. 557-597, 2020
The Collatz-Wielandt quotient for pairs of nonnegative operators
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