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 completely positive
operators. We give a characterization of minimal optimal solutions and
polynomially computable bounds on the Collatz-Wielandt quotient.Comment: 24 pages. To appear in Applications of Mathematics, ISSN 0862-794
