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When powers of a matrix coincide with its Hadamard powers

100%

Drnovšek R.

tom 3

nr 1

We characterize matrices whose powers coincide with their Hadamard powers.

Once more on positive commutators

100%

Drnovšek R.

Studia Mathematica

2012

tom 211

nr 3

241-245

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.

An irreducible semigroup of idempotents

100%

Drnovšek R.

Studia Mathematica

1997

tom 125

nr 1

97-99

We construct a semigroup of bounded idempotents with no nontrivial invariant closed subspace. This answers a question which was open for some time.

On operator bands

38%

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.