Subspaces Of Operators Research Articles

Triangulation of refined families

We prove the global triangulation conjecture for families of refined p -adic representations under a mild condition. That is, for a refined family, the associated family of (\varphi, \Gamma) -modules admits a global triangulation on a Zariski open and dense subspace of the base that contains all regular non-critical points. We also determine a large class of points which belongs to the locus of global triangulation. Furthermore, we prove that all the specializations of a refined family are trianguline. In the case of the Coleman–Mazur eigencurve, our results provide the key ingredient for showing its properness in a subsequent work [15].

A universal Riemannian foliated space

It is proved that the isometry classes of pointed connected complete Riemannian n-manifolds form a Polish space, M⁎∞(n), with the topology described by the C∞ convergence of manifolds. This space has a canonical partition into sets defined by varying the distinguished point into each manifold. The locally non-periodic manifolds define an open dense subspace M⁎,lnp∞(n)⊂M⁎∞(n), which becomes a C∞ foliated space with the restriction of the canonical partition. Its leaves without holonomy form the subspace M⁎,np∞(n)⊂M⁎,lnp∞(n) defined by the non-periodic manifolds. Moreover the leaves have a natural Riemannian structure so that M⁎,lnp∞(n) becomes a Riemannian foliated space, which is universal among all sequential Riemannian foliated spaces satisfying certain property called covering-determination. M⁎,lnp∞(n) is used to characterize the realization of complete connected Riemannian manifolds as dense leaves of covering-determined compact sequential Riemannian foliated spaces.

On hesitant multi-fuzzy soft topology

The aim of this paper is to introduce and study the concept of hesitant multi-fuzzy soft topological space and some of its structural properties, such as the neighbourhood of hesitant multi-fuzzy soft sets and interior hesitant multi-fuzzy soft sets, the hesitant multi-fuzzy soft basis and hesitant multi-fuzzy soft subspace topology. Additionally, we introduce and define the hesitant multi-fuzzy soft cover, hesitant multi-fuzzy soft open cover, and hesitant multi-fuzzy soft compactness, and some important results on them are presented.

Perturbations of invariant subspaces of operators with Hilbert–Schmidt Hermitian components

The paper deals with bounded non-selfadjoint operators having Hilbert–Schmidt imaginary Hermitian components. A perturbation bound for invariant subspaces is established. Our results can be considered as a particular generalization of the well-known Davis–Kahan sin θ-theorem for selfadjoint operators.

Quasi-range-compatible affine maps on large operator spaces

Let and be finite-dimensional vector spaces over an arbitrary field, and be a subset of the space of all linear maps from to . A map is called range-compatible when it satisfies for all ; it is called quasi-range-compatible when the condition is only assumed to apply to the operators whose range does not include a fixed one-dimensional linear subspace of . Among the range-compatible maps are the so-called local maps for fixed . Recently, the range-compatible group homomorphisms on were classified when is a linear subspace of small codimension in . In this work, we consider several variations of that problem: we investigate range-compatible affine maps on affine subspaces of linear operators; when is a linear subspace, we give the optimal bound on its codimension for all quasi-range-compatible homomorphisms on to be local. Finally, we give the optimal upper bound on the codimension of an affine subspace of for all quasi-range-compatible affine maps on it to be local.

Short-time existence for the second order renormalization group flow in general dimensions

We prove local existence for the second order Renormalization Group flow initial value problem on closed Riemannian manifolds (M, g) in general dimensions, for initial metrics whose sectional curvatures KP satisfy the condition 1 + αKP > 0, at all points p ∈ M and planes P ⊂ TpM . This extends results previously proven for two and three dimensions. The second order approximation of the Renormalization Group flow for the nonlinear sigma model of quantum field theory, which we label the RG-2 flow, is specified by the PDE system (1) ∂ ∂t g = −2 Rc− 2 Rm . Here g is a Riemannian metric, Rc is its Ricci curvature, Rmij = g ggRiklmRjpqn, and α is a positive parameter. We note that for our purposes here, α can assume any real value. For α = 0, this system (1) reduces to the Ricci flow. One can see that the sign of the right hand side, which is roughly 1+α×Curvature, should have an impact on the behavior of the flow, and this has been confirmed in various settings: in particular, the size of the term influences the parabolicity of the flow. Oliynyk has shown in [10] that on a two-dimensional manifold, if the Gaussian curvature K satisfies the condition 1 + αK > 0, then the flow is (weakly) parabolic; while if 1 + αK 0 is satisfied for all sectional curvatures KP . In this note we extend this curvature criterion for short-time existence for RG-2 flow to all dimensions, as first announced in [7]. Our main result is the following: Theorem 1. Let (M, g0) be a closed n-dimensional Riemannian manifold. If 1 + αKP > 0 for all sectional curvatures KP (g0), at all points p ∈M and planes P ⊂ TpM , then there exists a unique solution g(t) of the initial value problem ∂ ∂tg = −2 Rc− α 2 Rm , g(0) = g0, on some time interval [0, T ). Remark 2. In [10], Oliynyk finds open subspaces of the space of smooth metrics that are invariant under the two-dimensional RG-2 flow, and for which the flow remains parabolic (resp. backward parabolic). We are currently investigating this for general dimensions. Proof. To prove the theorem, we calculate the principal symbol of the DeTurck-modified version of RG-2 flow, which is generated by the PDE system (compare with (1) above) (2) ∂ ∂t gij = −2Rij + LWu,ggij − α 2 Rmij . Here Wu,g = −giju−1 jk gg(∇puql− 1 2∇lupq) is the standard vector field usually chosen to modify the Ricci flow into the related (parabolic) DeTurck version of Ricci flow, with u a fixed metric. Letting φt be the one-parameter family of diffeomorphisms generated by the vector field −Wu,g, then φt g is a solution of the RG-2 flow (see also [8]). As in the analogous Ricci flow case, if one can show (for a class of choices of the initial metric) that the PDE system (2) is parabolic, then short-time existence holds for the RG-2 flow (1) as well as for the DeTurck-modified flow (2). Date: January 7, 2014. KG is partially supported by the NSF under grant DGE-1144155. CG is partially supported by the Simons Foundation Collaboration Grant for Mathematicians 283083. JI is partially supported by the NSF under grant PHY-1306441 at the University of Oregon. He also wishes to thank the Mathematical Sciences Research Institute in Berkeley, California, for support under grant 0932078 000. Some of this work was carried out while JI was in residence at MSRI during the fall of 2013. 1 2 KARSTEN GIMRE, CHRISTINE GUENTHER, AND JAMES ISENBERG To calculate the symbol of the system (2), we first linearize the flow. For the first two terms of the right hand side of (2), this linearization effectively produces the Laplacian (see [4], or Theorem 2.1 in [5]). For the remaining term, Rm, it is useful to recall the formula for the variation of the Riemann curvature tensor with respect to the metric (see pg. 74 in [2]): [DRmg(h)] l ijk = [ ∂ ∂e Rm(g + eh) ∣∣∣∣ e=0 ]l

On Superactivation of Zero-Error Capacities and Reversibility of a Quantum Channel

We propose examples of low dimensional quantum channels demonstrating different forms of superactivation of one-shot zero-error capacities, in particular, the extreme superactivation (this complements the recent result of T.S.Cubitt and G.Smith). We also describe classes of quantum channels whose zero-error classical and quantum capacities cannot be superactivated. We consider implications of the superactivation of one-shot zero-error capacities to analysis of reversibility of a tensor-product channel with respect to families of pure states. Our approach based on the notions of complementary channel and of transitive subspace of operators makes it possible to study the superactivation effects for infinite-dimensional channels as well.

Non-meager free sets for meager relations on Polish spaces

of nowhere densesubsets of X;• is nowhere meager in X, if for any non-empty open set U⊂ Xthe intersection U∩Ais notmeager in X.It is clear that a subset A⊂ X of a Polish space X is nowhere meager if and only if Ais densein Xand contains no open meager subspace. By definition, cof(M) is the minimal cardinality of acollection X of meager subsets of the Baire space ω

Invariant subspaces of composition operators

The invariant subspace lattices of composition operators acting on H 2 , the Hilbert-Hardy space over the unit disc, are characterized in select cases. The lattice of all spaces left invariant by both a composition operator and the unilateral shift Mz (the multiplication operator induced by the co- ordinate function), is shown to be nontrivial and is completely described in particular cases. Given an analytic selfmap j of the unit disc, we prove that j has an angular derivative at some point on the unit circle if and only if Cj, the composition operator induced by j, maps certain subspaces in the invariant subspace lattice of Mz into other such spaces. A similar characterization of the existence of angular derivatives of j, this time in terms of Aj, the Aleksandrov operator induced by j, is obtained.

Reducing subspaces of multiplication operators with the symbol αz k + βw l on $$L_a^2 (\mathbb{D}^2 )$$

Multiplication operators defined on function spaces have been receiving enormous attention from both operator-theoretic and function-theoretic experts. One of the problems is to study reducing subspaces of them. The one-variable case has obtained fruitful remarkable results. However, little has been done in the multi-variable case. Under the setting of the Bergman space \(L_a^2 (\mathbb{D}^2 )\), this paper addresses those multiplication operators M p defined by special polynomials p, where p(z,w) = αz k +βw l , α, β ∈ ℂ. Those reducing subspaces of M p are completely determined.

Invariant subspaces of composition operators on a Hilbert space of Dirichlet series

In this paper, we study invariant subspaces of composition operators on the Hilbert space of Dirichlet series with square summable coefficients. The structure of invariant subspaces of a composition operator is characterized, and the strongly closed algebras generated by some composition operators with irrational symbols are shown to be reflexive. As an application, we provide a criterion for composition operators with certain symbols not to be algebraic.

Topological extensions with compact remainder

Let $\mathfrak{P}$ be a topological property. We study the relation between the order structure of the set of all $\mathfrak{P}$-extensions of a completely regular space $X$ with compact remainder (partially ordered by the standard partial order $\leq$) and the topology of certain subspaces of the outgrowth $\beta X\setminus X$. The cases when $\mathfrak{P}$ is either pseudocompactness or realcompactness are studied in more detail.

Существование гиперциклических подпространств у операторов Теплица

Существование гиперциклических подпространств у операторов Теплица

Operations on arc diagrams and degenerations for invariant subspaces of linear operators

We study geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are reductive algebraic groups acting on these varieties. We give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described.

ON HENSTROCK INTEGRALS OF INTERVAL-VALUED FUNCTIONS ON TIME SCALES

In this paper we introduce the interval-valued Hen- stock integral on time scales and investigate some properties of these integrals. 1. Introduction and preliminaries The Henstock integral for real functions was flrst deflned by Henstock (2) in 1963. The Henstock integral is more powerful and simpler than the Lebesgue, Wiener and Feynman integrals. The Henstock delta integral on time scales was introduced by Allan Peterson and Bevan Thompson (5) in 2006. In 2000, Congxin Wu and Zengtai Gong introduced the concept of the Henstock integral of interval-valued functions (6). In this paper we introduce the concept of the Henstock delta integral of interval-valued function on time scales and investigate some properties of the integral. A time scale T is a nonempty closed subset of real numberR with the subspace topology inherited from the standard topology ofR. For t 2 T we deflne the forward jump operator ae(t) = inffs 2 T : s > tg where inf ` = supfTg, while the backward jump operator ‰(t) = supfs 2 T : s t, we say that t is right-scattered, while if ‰(t) < t, we say that t is left-scattered. If ae(t) = t, we say that t is right-dense, while if ‰(t) = t, we say that t is left-dense. The forward graininess function (t) of t 2 T is deflned by (t) = ae(t) i t, whlie the backward graininess function (t) of t 2 T is deflned by (t) = t i ‰(t). For a;b 2 T we denote the closed interval (a;b)T = ft 2 T : atbg.

On a class of convolution operators on a finite interval

In this paper, we present criteria for the completeness of root subspaces of convolution operators on a finite interval with kernels equal to the Fourier transform of the ratio of two entire functions of exponential type from special classes. We describe a class of entire functions with a regular (in a certain sense) distribution of roots.

Localizing algebras and invariant subspaces

A theorem is provided about the existence of hyperinvariant subspaces for operators with a localizing subspace of extended eigenoperators. This theorem extends and unifies some previously known results of Scott Brown and Kim, Moore and Pearcy, and Lomonosov, Radjavi and Troitsky. Also, it is shown that the algebra L∞(μ) of all bounded measurable functions with respect to a finite measure μ is localizing on the Hilbert space L2(μ) if and only if the measure μ has an atom. Next, it is shown that the algebra H∞(D) of all bounded analytic multipliers on the unit disc fails to be localizing, both on the Bergman space A2(D) and on the Hardy space H2(D). Finally, several conditions are provided for the algebra generated by a diagonal operator on a Hilbert space to be localizing.

Arakelov–Parshin rigidity of towers of curve fibrations

Arakelov–Parshin rigidity is concerned with varieties mapping rigidly to the moduli stack $$\mathfrak {M}_h$$ of canonically polarized manifolds. Affirmative answer for any class of maps implies finiteness of the given class. This article studies Arakelov–Parshin rigidity on an open subspace of $$\mathfrak {M}_h$$ , on the locus $$\mathfrak {K}\mathfrak {F}_h$$ of iterated Kodaira fibrations. First, we prove rigidity for all complete curves mapping finitely onto $$\mathfrak {K}\mathfrak {F}_h$$ . Then, for generic affine curves mapping into $$\mathfrak {K}\mathfrak {F}_h$$ , rigidity is shown when $$\deg h =2$$ . The method used in the latter part is showing that the iterated Kodaira–Spencer map is injective.

On the reflexivity of subspaces of Toeplitz operators in simply connected regions

On the reflexivity of subspaces of Toeplitz operators in simply connected regions

On similarity and reducing subspaces of multiplication operator on Sobolev disk algebra

Let D be the unit disk and SA(D) denote the Sobolev disk algebra which consists of all analytic functions in the Sobolev space W2,2(D). In this note, we prove that Mzn is similar to ⨁1nMz on SA(D). Then using the matrix manipulations combined with operator theory methods, we characterize the reducing subspaces of Mzn on SA(D).