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Universal bounds for matrix semigroups

100%

Livshits L. , MacDonald G. , Radjavi H.

Studia Mathematica

2011

tom 203

nr 1

69-77

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.

Universal bounds for positive matrix semigroups

100%

Livshits L. , MacDonald G. , Marcoux L. , Radjavi H.

Studia Mathematica

2016

tom 232

nr 2

143-153

We show that any compact semigroup of positive n × n matrices is similar (via a positive diagonal similarity) to a semigroup bounded by √n. We give examples to show this bound is best possible. We also consider the effect of additional conditions on the semigroup and obtain improved bounds in some cases.

On operator bands

75%

Drnovšek R. , Livshits L. , MacDonald G. W. , Mathes B. , Radjavi H. , Šemrl P.

Studia Mathematica

2000

tom 139

nr 1

91-100

A multiplicative semigroup of idempotent operators is called an operator band. We prove that for each K>1 there exists an irreducible operator band on the Hilbert space $l^2$ which is norm-bounded by K. This implies that there exists an irreducible operator band on a Banach space such that each member has operator norm equal to 1. Given a positive integer r, we introduce a notion of weak r-transitivity of a set of bounded operators on a Banach space. We construct an operator band on $l^2$ that is weakly r-transitive and is not weakly (r+1)-transitive. We also study operator bands S satisfying a polynomial identity p(A, B) = 0 for all non-zero A,B ∈ S, where p is a given polynomial in two non-commuting variables. It turns out that the polynomial $p(A, B) = (A B - B A)^2$ has a special role in these considerations.