The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland

Applications of Mathematics (2020)

  • Volume: 65, Issue: 5, page 557-597
  • ISSN: 0862-7940

Abstract

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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.

How to cite

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Friedland, Shmuel. "The Collatz-Wielandt quotient for pairs of nonnegative operators." Applications of Mathematics 65.5 (2020): 557-597. <http://eudml.org/doc/297197>.

@article{Friedland2020,
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.},
author = {Friedland, Shmuel},
journal = {Applications of Mathematics},
keywords = {Perron-Frobenius theory; Collatz-Wielandt quotient; completely positive operator; commodity pricing; wireless network; quantum information theory},
language = {eng},
number = {5},
pages = {557-597},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The Collatz-Wielandt quotient for pairs of nonnegative operators},
url = {http://eudml.org/doc/297197},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Friedland, Shmuel
TI - The Collatz-Wielandt quotient for pairs of nonnegative operators
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 5
SP - 557
EP - 597
AB - 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.
LA - eng
KW - Perron-Frobenius theory; Collatz-Wielandt quotient; completely positive operator; commodity pricing; wireless network; quantum information theory
UR - http://eudml.org/doc/297197
ER -

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