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On operator bands

100%
Drnovšek R., Livshits L., MacDonald G. W., Mathes B., Radjavi H., Šemrl P.
Studia Mathematica
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2000
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tom 139
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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.
2
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An irreducible semigroup of idempotents

100%
Drnovšek R.
Studia Mathematica
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1997
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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.
3
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When powers of a matrix coincide with its Hadamard powers

100%
Drnovšek R.
Special Matrices
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2015
|
tom 3
|
nr 1
EN
We characterize matrices whose powers coincide with their Hadamard powers.
4
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Invo-regular unital rings

75%
Danchev P. V.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2018
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tom 72
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nr 1
EN
It was asked by Nicholson (Comm. Algebra, 1999) whether or not unit-regular rings are themselves strongly clean. Although they are clean as proved by Camillo-Khurana (Comm. Algebra, 2001), recently Nielsen and Ster showed in Trans. Amer. Math. Soc., 2018 that there exists a unit-regular ring which is not strongly clean. However, we define here a proper subclass of rings of the class of unit-regular rings, called invo-regular rings, and establish that they are strongly clean. Interestingly, without any concrete indications a priori, these rings are manifestly even commutative invo-clean as defined by the author in Commun. Korean Math. Soc., 2017.
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