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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.

On operator bands

100%

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.

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.

Dynamic projection operator method in the theory of hyperbolic systems of partial differential equations with variable coefficients

100%

Leble S. , Vereshchagina I.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2017

tom Vol. 21, No 2

109--120

We consider a generalization of the projection operator method for the case of the Cauchy problem in 1D space for systems of evolution differential equations of first order with variable coefficients. It is supposed that the dependence of coefficients on the only variable χ is weak, that is described by the introduction of a small parameter. Such problem corresponds, for example, to the case of wave propagation in a weakly inhomogeneous medium. As an example, we specify the problem to adiabatic acoustics in waveguides with a variable cross-section. Projection operators are constructed for the Cauchy problem to fix unidirectional modes. The method of successive approximations (perturbation theory) is developed and based on the pseudodifferential operators theory. The application of projection operators adapted for the case under consideration allows deriving approximate evolution equations corresponding to the separated directed waves.