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An irreducible semigroup of idempotents

100%
Drnovšek R.
Studia Mathematica
|
1997
|
tom 125
|
nr 1
97-99
EN
We construct a semigroup of bounded idempotents with no nontrivial invariant closed subspace. This answers a question which was open for some time.
2
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When powers of a matrix coincide with its Hadamard powers

100%
Drnovšek R.
Special Matrices
|
2015
|
tom 3
|
nr 1
EN
We characterize matrices whose powers coincide with their Hadamard powers.
3
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Once more on positive commutators

100%
Drnovšek R.
Studia Mathematica
|
2012
|
tom 211
|
nr 3
241-245
EN
Let A and B be bounded operators on a Banach lattice E such that the commutator C = AB - BA and the product BA are positive operators. If the product AB is a power-compact operator, then C is a quasi-nilpotent operator having a triangularizing chain of closed ideals of E. This answers an open question posed by Bračič et al. [Positivity 14 (2010)], where the study of positive commutators of positive operators was initiated.
4
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On operator bands

39%
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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