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In this article we study a natural weakening – which we refer to as paratransitivity – of the well-known notion of transitivity of an algebra AA of linear operators acting on a finite-dimensional vector space VV. Given positive integers k and m, we shall say that such an algebra AA is (k,m)(k,m)...
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In this paper, the authors continue the analysis, first undertaken in Livshits et al. (2013) [2], of algebras AA of linear transformations on an n-dimensional complex vector space VV which have the property that if WW is a k-dimensional subspace of VV, then the image of WW under the action of the al...
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The following questions are studied: Under what conditions does the existence of a (nonzero) fixed point for every member of a semigroup of matrices imply a common fixed point for the entire semigroup? What is the smallest number k such that the existence of a common fixed point for every k members ...
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We focus on matrix semigroups (and algebras) on which rank is commutable [rank(AB) = rank(BA)]. It is shown that in a number of cases (for example, in dimensions less than 6), but not always, commutativity of rank entails permutability of rank [rank(A(1)A(2)...A(n)) = rank(A(sigma(1))A(sigma(2))... ...
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A description of the lattice of invariant subspaces is provided for multiplicative semigroups S of bounded operators on L-p(X, mu) which are closed under multiplication on the left or right by bounded multiplication operators. Applications are then given to semigroups of positive quasinilpotent oper...
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We construct an irreducible multiplicative semigroup of non-negative square-zero operators acting on L-P[0, 1), for 1 less than or equal to p < infinity.
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We investigate the structure of (minimal) n-transitive semigroups of matrices of rank at most n in M-k(F), by considering an equivalent problem regarding certain families of (n - k)-dimensional subspaces of F-k. (C) 2001 Elsevier Science Inc. All rights reserved.
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It is Shown that a nest in a Hilbert space H is the lattice of closed invariant subspaces of a band algebra in B(H) (i.e. an algebra generated by a semigroup of idempotent operators) if and only if all finite-dimensional atoms of the nest have dimension 1. A canonical operator matrix form for operat...
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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 ...
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This paper deals with semigroups of linear transformations which act transitively on a finite-dimensional vector space. An explicit canonical form is obtained for the semigroups which lack proper transitive left ideals. The class of such semigroups can be considered to be an extention of the class o...
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Any pure operator band can be expanded so that each component of the band is reflexive.
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We study the existence of common invariant subspaces for semigroups of idempotent operators. It is known that in finite dimensions every such semigroup is simultaneously triangularizable. The question; of the existence of even one non-trivial invariant subspace is still open in infinite dimensions. ...
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The distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k x k matrices with the usual operator norm, is shown to be sec(pi/(k+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the ...
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