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Universal bounds for positive matrix semigroups

100%
Livshits L., MacDonald G., Marcoux L., Radjavi H.
Studia Mathematica
|
2016
|
tom 232
|
nr 2
143-153
EN
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.
2
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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
EN
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.
3
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On operator bands

76%
Drnovšek R., Livshits L., MacDonald G. W., Mathes B., Radjavi H., Šemrl P.
Studia Mathematica
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2000
|
tom 139
|
nr 1
91-100
EN
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.
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