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Norm estimates for solutions of matrix equations AX-XB=C and X-AXB=C

100%
Gil' M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2014
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tom 34
|
nr 2
191-206
EN
Let A, B and C be matrices. We consider the matrix equations Y-AYB=C and AX-XB=C. Sharp norm estimates for solutions of these equations are derived. By these estimates a bound for the distance between invariant subspaces of matrices is obtained.
2
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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
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.
3
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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.
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