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The Collatz-Wielandt quotient for pairs of nonnegative operators

Shmuel Friedland (2020)

Applications of Mathematics

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...

Universal bounds for matrix semigroups

Leo Livshits, Gordon MacDonald, Heydar Radjavi (2011)

Studia Mathematica

We show that any compact semigroup of n × n matrices is similar to a semigroup bounded by √n. We give examples to show that this bound is best possible and consider the effect of the minimal rank of matrices in the semigroup on this bound.

Young's (in)equality for compact operators

Gabriel Larotonda (2016)

Studia Mathematica

If a,b are n × n matrices, T. Ando proved that Young’s inequality is valid for their singular values: if p > 1 and 1/p + 1/q = 1, then λ k ( | a b * | ) ≤ λ k ( 1 / p | a | p + 1 / q | b | q ) for all k. Later, this result was extended to the singular values of a pair of compact operators acting on a Hilbert space by J. Erlijman, D. R. Farenick and R. Zeng. In this paper we prove that if a,b are compact operators, then equality holds in Young’s inequality if and only if | a | p = | b | q .

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