IRREDUCIBLE ACTIONS AND COMPRESSIBLE MODULES

Any finite set of linear operators on an algebra A yields an operator algebra B and a module structure on A, whose endomorphism ring is isomorphic to a subring AB of certain invariant elements of A. We show that if A is a critically compressible left B-module, then the dimension of its self-injective hull Â over the ring of fractions of AB is bounded by the uniform dimension of A and the number of linear operators generating B. This extends a known result on irreducible Hopf actions and applies in particular to weak Hopf action. Furthermore we prove necessary and sufficient conditions for an algebra A to be critically compressible in the case of group actions, group gradings and Lie actions.

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The Zak Transform on Gelfand–Shilov and Modulation Spaces with Applications to Operator Theory

Complex Analysis and Operator Theory ◽

10.1007/s11785-020-01039-6 ◽

2020 ◽

Vol 15 (1) ◽

Author(s):

Joachim Toft

Keyword(s):

Operator Theory ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Linear Operators ◽

Modulation Spaces ◽

Zak Transform ◽

Periodic Functions ◽

Distribution Spaces ◽

Necessary And Sufficient

AbstractWe characterize Gelfand–Shilov spaces, their distribution spaces and modulation spaces in terms of estimates of their Zak transforms. We use these results for general investigations of quasi-periodic functions and distributions. We also establish necessary and sufficient conditions for linear operators in order for these operators should be conjugations by Zak transforms.

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Rado's theorem for polymatroids

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100051677 ◽

1975 ◽

Vol 78 (2) ◽

pp. 263-281 ◽

Cited By ~ 43

Author(s):

Colin J. H. McDiarmid

Keyword(s):

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Rank Function ◽

Fundamental Importance ◽

Matroid Theory ◽

Continuous Analogue ◽

Finite Set ◽

Necessary And Sufficient ◽

Transversal Theory

The theorem of R. Rado (12) to which I refer by the name ‘Rado's theorem for matroids’ gives necessary and sufficient conditions for a family of subsets of a finite set Y to have a transversal independent in a given matroid on Y. This theorem is of fundamental importance in both transversal theory and matroid theory (see, for example, (11)). In (3) J. Edmonds introduced and studied ‘polymatroids’ as a sort of continuous analogue of a matroid. I start this paper with a brief introduction to polymatroids, emphasizing the role of the ‘ground-set rank function’. The main result is an analogue for polymatroids of Rado's theorem for matroids, which I call not unnaturally ‘Rado's theorem for polymatroids’.

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Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

Canadian Mathematical Bulletin ◽

10.4153/cmb-1988-053-3 ◽

1988 ◽

Vol 31 (3) ◽

pp. 374-379 ◽

Cited By ~ 1

Author(s):

Kenneth G. Wolfson

Keyword(s):

Left Ideal ◽

Endomorphism Ring ◽

Finite Rank ◽

Sufficient Conditions ◽

Free Module ◽

Necessary And Sufficient Conditions ◽

Finitely Generated ◽

Ore Domain ◽

Necessary And Sufficient

AbstractA prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

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On the invertibility of length two elementary operators

Electronic Journal of Linear Algebra ◽

10.13001/1081-3810.3124 ◽

2015 ◽

Vol 30 ◽

pp. 916-913

Author(s):

Janko Bracic ◽

Nadia Boudi

Keyword(s):

Banach Space ◽

Sufficient Conditions ◽

Complex Banach Space ◽

Linear Operators ◽

Elementary Operator ◽

Bounded Linear Operators ◽

Existence Of A Solution ◽

Bounded Linear ◽

Elementary Operators ◽

Necessary And Sufficient

Let X be a complex Banach space and L(X) be the algebra of all bounded linear operators on X. For a given elementary operator P of length 2 on L(X), we determine necessary and sufficient conditions for the existence of a solution of the equation YP=0 in the algebra of all elementary operators on L(X). Our approach allows us to characterize some invertible elementary operators of length 2 whose inverses are elementary operators.

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Square roots in finite full transformation semigroups

Glasgow Mathematical Journal ◽

10.1017/s0017089500004912 ◽

1982 ◽

Vol 23 (2) ◽

pp. 137-149 ◽

Cited By ~ 5

Author(s):

Mary Snowden ◽

J. M. Howie

Keyword(s):

Transformation Semigroup ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Necessary Condition ◽

Square Root ◽

Transformation Semigroups ◽

Full Transformation ◽

Square Roots ◽

Finite Set ◽

Necessary And Sufficient

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

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Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces

Canadian Mathematical Bulletin ◽

10.4153/cmb-2017-057-6 ◽

2018 ◽

Vol 61 (4) ◽

pp. 717-737 ◽

Cited By ~ 1

Author(s):

Shangquan Bu ◽

Gang Cai

Keyword(s):

Sufficient Conditions ◽

Complex Banach Space ◽

Second Order ◽

Linear Operators ◽

Bounded Linear ◽

Order Degenerate ◽

Degenerate Differential Equations ◽

Necessary And Sufficient ◽

Equations With Delay ◽

Closed Linear Operators

AbstractWe give necessary and sufficient conditions of the Lp-well-posedness (resp. -wellposedness) for the second order degenerate differential equation with finite delayswith periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′ (0) = (Mu)′ (2π), where A, B, and M are closed linear operators on a complex Banach space X satisfying D(A) ∩ D(B) ⊂ D(M), F and G are bounded linear operators from into X.

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Projections and embeddings of locally convex operator spaces and their duals

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100062228 ◽

1984 ◽

Vol 96 (2) ◽

pp. 321-323 ◽

Cited By ~ 3

Author(s):

Jan H. Fourie ◽

William H. Ruckle

Keyword(s):

Dual Space ◽

Sufficient Conditions ◽

Linear Operators ◽

Locally Convex Spaces ◽

Continuous Linear ◽

Operator Spaces ◽

Complemented Subspace ◽

Necessary And Sufficient ◽

Locally Convex ◽

Continuous Linear Operators

AbstractLet E, F be Hausdorff locally convex spaces. In this note we consider conditions on E and F such that the dual space of the space Kb (E, F) (of quasi-compact operators) is a complemented subspace of the dual space of Lb (E, F) (of continuous linear operators). We obtain necessary and sufficient conditions for Lb(E, F) to be semi-reflexive.

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New Results on Markov Moment Problem

International Journal of Analysis ◽

10.1155/2013/901318 ◽

2013 ◽

Vol 2013 ◽

pp. 1-17 ◽

Cited By ~ 2

Author(s):

Octav Olteanu

Keyword(s):

Moment Problem ◽

Sufficient Conditions ◽

Linear Operators ◽

Moment Problems ◽

Finite Dimensional ◽

Integrable Functions ◽

Truncated Moment Problem ◽

Markov Moment ◽

The Moment ◽

Necessary And Sufficient

The present work deals with the existence of the solutions of some Markov moment problems. Necessary conditions, as well as necessary and sufficient conditions, are discussed. One recalls the background containing applications of extension results of linear operators with two constraints to the moment problem and approximation by polynomials on unbounded closed finite-dimensional subsets. Two domain spaces are considered: spaces of absolute integrable functions and spaces of analytic functions. Operator valued moment problems are solved in the latter case. In this paper, there is a section that contains new results, making the connection to some other topics: bang-bang principle, truncated moment problem, weak compactness, and convergence. Finally, a general independent statement with respect to polynomials is discussed.

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Local conditions for the stochastic comparison of particle systems

Advances in Applied Probability ◽

10.1239/aap/1103662966 ◽

2004 ◽

Vol 36 (4) ◽

pp. 1252-1277 ◽

Cited By ~ 1

Author(s):

Rosario Delgado ◽

F. Javier López ◽

Gerardo Sanz

Keyword(s):

Queueing Networks ◽

Interacting Particle Systems ◽

Sufficient Conditions ◽

Explicit Construction ◽

Particle Systems ◽

Stochastic Comparison ◽

Local Conditions ◽

Jackson Networks ◽

Finite Set ◽

Necessary And Sufficient

We study the stochastic comparison of interacting particle systems where the state space of each particle is a finite set endowed with a partial order, and several particles may change their value at a time. For these processes we give local conditions, on the rates of change, that assure their comparability. We also analyze the case where one of the processes does not have any changes that involve several particles, and obtain necessary and sufficient conditions for their comparability. The proofs are based on the explicit construction of an order-preserving Markovian coupling. We show the applicability of our results by studying the stochastic comparison of infinite-station Jackson networks and batch-arrival, batch-service, and assemble-transfer queueing networks.

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Eigenvalue problems for the wave equation with strong damping

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023805 ◽

1997 ◽

Vol 127 (4) ◽

pp. 755-771 ◽

Cited By ~ 4

Author(s):

Pedro Freitas

Keyword(s):

Stationary Solution ◽

Wave Equations ◽

Eigenvalue Problems ◽

Sufficient Conditions ◽

Stationary Solutions ◽

Linear Operators ◽

Stability And Instability ◽

The Stability ◽

Necessary And Sufficient ◽

Strongly Damped

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

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