Subspaces Of Operators Research Articles

On Semi-?-Connected Subspace

In this paper, the concept of semi-?-open set will be used to define a new kind of strongly connectedness on a topological subspace namely "semi-?-connectedness". Moreover, we prove that semi-?-connectedness property is a topological property and give an example to show that semi-?-connectedness property is not a hereditary property. Also, we prove thate semi-?-irresolute image of a semi-?-connected space is a semi-?-connected space.

A note on a Halmos problem

AbstractWe address the existence of non‐trivial closed invariant subspaces of operators on Banach spaces whenever their square have or, more generally, whether there exists a polynomial with such that the lattice of invariant subspaces of is non‐trivial. In the Hilbert space setting, the ‐problem was posed by Halmos in the seventies and in 2007, Foias, Jung, Ko and Pearcy conjectured it could be equivalent to the Invariant Subspace Problem.

Maldistributed sequences in metric spaces

Let a kind of measure μ be given on the set of all positive integers N. By a μ-maldistributed sequence in a metric space (X,d) we mean any sequence possessing μ-extremely many points in each nonempty open subspace of (X,d). Let s(X) denote the metric space of all sequences in (X,d) endowed with the Fréchet metric. In this paper we study topological properties of the set of all maldistributed sequences in X as a subspace s(X). We show that if (X,d) is separable then this set is residual for a large class of commonly used measures μ on N.

Hyperinvariant subspaces for operators intertwined with weighted shift operators

Hyperinvariant subspaces for operators intertwined with weighted shift operators

Analytic representations of backward shifts on weighted Bergman and Dirichlet spaces

The backward shifts (the adjoints of multiplication by z) on weighted Bergman spaces of the unit disk are important in operator theory. For example, Agler [1] shows their restrictions to invariant subspaces of vector-valued weighted Bergman spaces are model operators for hypercontractions. We present analytic representations of backward shifts on weighted Bergman spaces and Dirichlet type spaces. Since these analytic representations are not space specific, it is interesting to study these operators on Hilbert spaces of holomorprhic functions on the unit disk. We initiate this study by developing several properties of these operators. In particular, we obtain simple proofs and extend the invariant subspace theorem of shift plus Volterra operator on the Hardy space by Čučković and Paudyal [16]. Furthermore, we describe invariant subspaces of shift plus Volterra operator on the Bergman space by relating them to the invariant subspaces of the Dirichlet shift. Analytic representations of the tuple of backward shifts on weighted Bergman spaces of the unit ball are also discussed.

Thurston’s asymmetric metric on the space of singular flat metrics with a fixed quadrangulation

Consider a compact surface equipped with a fixed quadrangulation. One may identify each quadrangle on the surface with a Euclidean rectangle to obtain a singular flat metric on the surface with conical singularities. We call such a singular flat metric a rectangular structure. We study a metric on the space of unit area rectangular structures which is analogous to Thurston’s asymmetric metric on the Teichmüller space of a surface of finite type. We prove that the distance between two rectangular structures is equal to the logarithm of the maximum of ratios of edges of these rectangular structures. We give a sufficient condition for a path between two points of the this Teichmüller space to be geodesic and we prove that any two points of this space can be joined by a geodesic. We also prove that this metric is Finsler and give a formula for the infinitesimal weak norm on the tangent space of each point. We identify the space of unit area rectangular structures with a submanifold of a Euclidean space and we show that the subspace topology and the topology induced by the metric we introduced coincide. We show that the space of unit area rectangular structures on a surface with a fixed quadrangulation is in general not complete.

Reflexive extended locally convex spaces

For an extended locally convex space (elcs) (X,τ), the authors in [9] studied the topology τucb of uniform convergence on bounded subsets of (X,τ). In the present paper, we use the topology τucb to explore the reflexive property of extended locally convex spaces. As a main result, we show that an elcs is (semi) reflexive if and only if any of its open subspaces is (semi) reflexive. For an extended normed space, we show that reflexivity is a three-space property.

The Space of Fuzzy Minimal Prime Filters In an Almost Distributive Lattice

In this paper, we establish and describe the notions of fuzzy minimal prime filters of an Almost Distributive Lattice (ADL). We prove that a fuzzy filter is the point wise infimum of all minimal prime fuzzy filters (all fuzzy minimal prime filters) of an ADL. Mainly, we introduce the topological space on the set of all fuzzy minimal prime filters of an ADL (denoted by M(R)). For an ADL R, we prove that an open set M(R) is a base for the topology and it constitutes a base for the subspace topology on a closed set N and they are the only compact open sets.

A Stronger Novel Form of Soft Semi-Open Set

In this paper, we introduce the concept of a soft [Formula: see text]-open set. We show that the class of soft [Formula: see text]-open sets lies strictly between the classes of [Formula: see text]-open sets (respectively, soft [Formula: see text]-open) and soft semi-open sets. Additionally, we provide several adequate criteria for the equivalence between soft [Formula: see text]-open sets and each of [Formula: see text]-open sets and soft semi-open sets. Moreover, we demonstrate that the family of soft [Formula: see text]-open sets is a supra soft topology. We study the relationships between soft [Formula: see text]-open sets and other types of soft sets such as soft open, soft pre-open, soft [Formula: see text]-open, soft [Formula: see text]-open, soft [Formula: see text]-open, soft [Formula: see text]-open and soft [Formula: see text]-open. Also, we study the behavior of soft [Formula: see text]-open sets under some soft operators. Finally, we clarify the correspondence between the classes of soft [Formula: see text]-open sets in a soft topological space and in its soft topological subspaces.

Zariski topology on the secondary-like spectrum of a module

Abstract Let ℜ \Re be a commutative ring with unity and ℑ \Im be a left ℜ \Re -module. We define the secondary-like spectrum of ℑ \Im to be the set of all secondary submodules K K of ℑ \Im such that the annihilator of the socle of K K is the radical of the annihilator of K K , and we denote it by Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) . In this study, we introduce a topology on Spec L ( ℑ ) {{\rm{Spec}}}^{L}\left(\Im ) having the Zariski topology on the second spectrum Spec s ( ℑ ) {{\rm{Spec}}}^{s}\left(\Im ) as a subspace topology and study several topological structures of this topology.

On isotropic subspaces of operators in II1 factors

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we firstly prove that given an X∈M, then there is a decomposition of the identity into N∈N⋃{+∞} mutually orthogonal nonzero projections {En}n=1N⊆M, I=∑n=1NEn, such that EnXEn=τ(X)En for all n=1,⋯,N. Secondly, we show that if R is the hyperfinite II1 factor and (Rω,τω) is the ultrapower of R, then for any X∈Rω, there is a finite family of mutually orthogonal nonzero projections {Ei}i=1N in Rω such that ∑i=1NEi=I and EnXEn=τ(X)En, for all n=1,2,⋯,N. Equivalently, there is a unitary operator U∈Rω such that 1N∑i=1N(U⁎)iXUi=τω(X)I.

Constructing the Oseledets decomposition with subspace growth estimates

The semi-invertible version of Oseledets’ multiplicative ergodic theorem providing a decomposition of the underlying state space of a random linear dynamical system into fast and slow spaces is deduced for a strongly measurable cocycle on a separable Banach space. This work is a much shorter means of obtaining this general version of the theorem, using measurable growth estimates on subspaces for linear operators combined with a modified version of Kingman’s subadditive ergodic theorem.

Covering properties of $ C_{p}\left(Y|X\right) $

<abstract><p>Let $ X $ be an infinite Tychonoff space, and $ Y $ be a topological subspace of $ X $. In this paper, we study some covering properties of the subspace $ C_{p}\left(Y|X\right) $ of $ C_{p}\left(Y\right) $ consisting of those functions $ f\in C\left(Y\right) $ which admit a continuous extension to $ X $ equipped with the relative topology of $ C_{p}\left(Y\right) $. Among other results, we show that $ \left(i\right) $ $ C_{p}(Y|X) $ has a fundamental bounded resolution if and only if $ Y $ is countable; when $ X $ is realcompact and $ Y $ is closed in $ X $, we have $ \left(ii\right) $ if $ C_{p}(Y|X) $ admits a resolution of convex compact sets that swallows the local null sequences in $ C_{p}(Y|X) $, then $ Y $ is countable and discrete; $ \left(iii\right) $ if $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets, then $ Y $ is also countable and discrete, and, as a corollary, we deduce that $ C_{p}(Y|X) $ admits a compact resolution that swallows the compact sets if and only if $ C_{p}(Y|X) $ is a Polish space. We also prove that $ \left(iv\right) $ for a metrizable space $ X $, $ C_{p}\left(X\right) $ is a quasi-$ \left(LB\right) $-space if and only if $ X $ is $ \sigma $-compact, and hence for a subspace $ Y $ of $ X $, the space $ C_{p}\left(Y|X\right) $ is a quasi-$ \left(LB\right) $-space. We include some examples and observations that answer natural questions raised in this paper.</p></abstract>

Topologies on unparameterised path space

The signature of a path, introduced by K.T. Chen [10] in 1954, has been extensively studied in recent years. The fundamental 2010 paper [20] of Hambly and Lyons showed that the signature is an injective function on the space of continuous, finite-variation paths up to a general notion of reparameterisation called tree-like equivalence. This result has been extended to geometric rough paths by Boedihardjo et al. [5]. More recently, the approximation theory of the signature has been widely used in the literature in applications. The archetypal instance of these results, see e.g. [24], guarantees uniform approximation, on compact sets, of a continuous function by a linear functional on the (extended) tensor algebra acting on the signature.In this paper we study in detail, and for the first time, the properties of three natural candidate topologies on the set of unparameterised paths, i.e. the tree-like equivalence classes. These are obtained by privileging different properties of the signature and are: (1) the product topology, obtained by equipping the range of the signature with the (subspace topology of the) product topology in the extended tensor algebra and then requiring S to be an embedding, (2) the quotient topology derived from the 1-variation topology on the underlying path space, and (3) the metric topology associated to d([γ],[σ]):=||γ⁎−σ⁎||1 using the (constant-speed) tree-reduced representatives γ⁎ and σ⁎ of the respective equivalence classes. We evaluate these spaces from the point of view of their suitability when it comes to studying (probability) measures on them. We prove that the respective collections of open sets are ordered by strict inclusion, (1) being the weakest and (3) the strongest. Our other conclusions can be summarised as follows. All three topological spaces are separable and Hausdorff, (1) being both metrisable and σ-compact, but not a Baire space and hence being neither Polish nor locally compact. The completion of (1), in any metric inducing the product topology, is the subspace G⁎ of group-like elements. The quotient topology (2) is not metrisable and the metric d is not complete. We also discuss some open problems related to these spaces.We consider finally the implications of the selection of the topology for uniform approximation results involving the signature. A stereotypical model for a continuous function on (unparameterised) path space is the solution of a controlled differential equation. We thus prove, for a broad class of these equations, well-definedness and measurability of the (fixed-time) solution map with respect to the Borel sigma-algebra of each topology. Under stronger regularity assumptions, we further show continuity of this same map on explicit compact subsets of the product topology (1). We relate these results to the expected signature model of [24].

Invariant subspaces of operators via Berezin symbols and Duhamel product

Invariant subspaces of operators via Berezin symbols and Duhamel product

The local properties in topological groups and related concepts and questions

The next general question is considered below: when certain open subspaces of topological groups are homeomorphic to some topological group? A version of this question has been posed in [4] (open problem 1.4.1). In it, the complements to the elements in any zero-dimensional topological group were considered. A result below in this direction is Theorem 3.1 saying that if G is a σ-compact topological group and F is a nonempty compact subspace of G such that the complement G∖F is homeomorphic to some topological group, then F is a Gδ-subset of G (and therefore, by a remarkable theorem of M.M. Choban [6], F is a dyadic compactum). Another conclusion in this theorem says that if, moreover, the tightness of F is countable (in particular, if F is first countable), or if F is topologically linearly orderable, then G has a countable network. Lemma 4.1 says that if X is a Lindelöf homogeneous space with countable Souslin number and X is puncturable at some point a, then a is a Gδ-point in X, the subspace Y=X∖{a} is Lindelöf, and X is submetrizable. It follows from this that every Lindelöf topological field with countable Souslin number is submetrizable (its topology contains a metrizable topology). A natural Abelian topological group with few open subspaces that are homeomorphic to topological groups is identified in Example 3.2.

Constrained deformations of positive scalar curvature metrics, II

AbstractWe prove that various spaces of constrained positive scalar curvature metrics on compact three‐manifolds with boundary, when not empty, are contractible. The constraints we mostly focus on are given in terms of local conditions on the mean curvature of the boundary, and our treatment includes both the mean‐convex and the minimal case. We then discuss the implications of these results on the topology of different subspaces of asymptotically flat initial data sets for the Einstein field equations in general relativity.

A variation of the class of statisticalγcovers

AbstractIn this article, we introduce s-s-γ\gammacover using the notion of star operator, which is an extension of the previous results on s-γ\gammacovers. We also define s-s-dense set to address the issue that a statistical dense subset of an s-s-γ\gammacover is not an s-s-γ\gammacover. Furthermore, we study some results on s-γ\gammacovers in subspace topology.

Polish topologies on endomorphism monoids of relational structures

In this paper we present general techniques for characterising minimal and maximal semigroup topologies on the endomorphism monoid End(A) of a countable relational structure A. As applications, we show that the endomorphism monoids of several well-known relational structures, including the random graph, the random directed graph, and the random partial order, possess a unique Polish semigroup topology. In every case this unique topology is the subspace topology induced by the usual topology on the Baire space NN. We also show that many of these structures have the property that every homomorphism from their endomorphism monoid to a second countable topological semigroup is continuous; referred to as automatic continuity. Many of the results about endomorphism monoids are extended to clones of polymorphisms on the same structures.

Representation theorems for regular operators

AbstractWe elaborate, strengthen, and generalize known representation theorems by different authors for regular operators on vector and Banach lattices. Our main result asserts, in particular, that every regular linear operator T acting from a vector lattice E with the principal projection property to a Dedekind complete vector lattice F, which is an ideal of some order continuous Banach lattice G, admits a unique representation , where is the sum of an absolutely order summable family of disjointness preserving operators and is an order narrow (= diffuse) operator. Our main contribution is waiver of the order continuity assumption on T. In proofs, we use new techniques that allow obtaining more general results for a wider class of orthogonally additive operators, which has somewhat different order structure than the linear subspace of linear operators.