Subspaces Of Operators Research Articles

On invariant subspaces of operators in the class [formula omitted

In this paper, we consider invariant subspaces of operators in the class θ, which is the set of operators T such that T∗T and T+T∗ commute. It is shown that every operator in the class θ such that the outer boundary of its spectrum is the outer boundary of a Carathéodory domain has a nontrivial invariant subspace. We also give a family of operators in the class θ which are reductive, i.e., their invariant subspaces are reducing. In addition, we give a condition on spectra of operators in the class θ which gives some information about invariant subspaces.

Fractal topology foundations

In this paper, we introduce the foundation of a fractal topological space constructed via a family of nested topological spaces endowed with subspace topologies, where the number of topological spaces involved in this family is related to the appearance of new structures on it. The greater the number of topological spaces we use, the stronger the subspace topologies we obtain. The fractal manifold model is brought up as an illustration of space that is locally homeomorphic to the fractal topological space.

Spaces of remote points

Given a Tychonoff space X, let ϱ(X) be the set of remote points of X. We view ϱ(X) as a topological space. In this paper we assume that X is metrizable and ask for conditions on Y so that ϱ(X) is homeomorphic to ϱ(Y). This question has been studied before by R.G. Woods and C. Gates. We give some results of the following type: if X has topological property P and ϱ(X) is homeomorphic to ϱ(Y), then Y also has P. We also characterize the remote points of the rationals and irrationals up to some restrictions. Further, we show that ϱ(X) and ϱ(Y) have open dense homeomorphic subspaces if X and Y are both nowhere locally compact, completely metrizable and share the same cellular type, a cardinal invariant we define.

On the existence of maximal semidefinite invariant subspaces for J-dissipative operators

For a certain class of operators we present some necessary and sufficient conditions for a J-dissipative operator in a Kreĭn space to have maximal semidefinite invariant subspaces. We investigate the semigroup properties of restrictions of the operator to these invariant subspaces. These results are applied to the case when the operator admits a matrix representation with respect to the canonical decomposition of the space. The main conditions are formulated in terms of interpolation theory for Banach spaces.Bibliography: 25 titles.

The Proper Elements and Simple Invariant Subspaces

A proper element of $X$ is a triple $(\lambda$, $L$, $A)$ composed by an eigenvalue $\lambda$, an invariant subspace of an operator $A$ in $B(X)$ generated by one eigenvector of $\lambda$ and the operator $A$. For $ (\lambda_0, L_0,A_0)\in Eig (X)$, where $L_0=\mathcal{L} (\{x_0\} )$, the operator $A_0$ induces an operator $\widehat{A_0}$ from the quotient $X/L_0$ into itself, i.e. $\widehat{A_0}(x+L_0)=A_0(x)+L_0$. In paper we show that $\lambda_0$ is a simple pole of $A_0$ if and only if $\lambda_0\notin\sigma (\widehat{A_0})$. Follow this concept we can define simple invariant subspaces of linear operator $T$ like invariant subspace $E$ such that $\sigma (T_E)\cap\sigma (\widehat{T_E})=\emptyset$, where $T_{E}:E\to E$ is the restriction of $T$ on $E$, $\widehat{T_E}$ is the operator $\widehat{T_E}(\pi (y))=\pi (T(y))$ on the quotient space $X/E$ and $\pi$ is the natural homoeomorphism between $X$ and $X/E$. Also, we give some properties of stability of simple invariant subspaces.

Illumination Invariant Facial Pose Classification

this paper, we compared the performance of various combinations of edge operators and linear subspace methods to determine the best combination for pose classification. To evaluate the performance, we have carried out experiments on CMU-PIE database which contains images with wide variation in illumination and pose. We found that the performance of pose classification depends on the choice of edge operator and linear subspace method. The best classification accuracy is obtained with Prewitt edge operator and Eigenfeature regularization method. In order to handle illumination variation, we used adaptive histogram equalization as a preprocessing step resulting into significant improvement in performance except for Roberts operator. General Terms Pattern Recognition, Classification. KeywordsClassification, Edge detection, Linear Subspace Methods.

Continuous extensions of functions defined on subsets of products

A subset Y of a space X is Gδ-dense if it intersects every nonempty Gδ-set. The Gδ-closure of Y in X is the largest subspace of X in which Y is Gδ-dense.The space X has a regularGδ-diagonal if the diagonal of X is the intersection of countably many regular-closed subsets of X×X.Consider now these results: (a) (N. Noble, 1972 [18]) every Gδ-dense subspace in a product of separable metric spaces is C-embedded; (b) (M. Ulmer, 1970 [22], 1973 [23]) every Σ-product in a product of first-countable spaces is C-embedded; (c) (R. Pol and E. Pol, 1976 [20], also A.V. Arhangelʼskiĭ, 2000 [3]; as corollaries of more general theorems), every dense subset of a product of completely regular, first-countable spaces is C-embedded in its Gδ-closure.The present authorsʼ Theorem 3.10 concerns the continuous extension of functions defined on subsets of product spaces with the κ-box topology. Here is the case κ=ω of Theorem 3.10, which simultaneously generalizes the above-mentioned results. TheoremLet{Xi:i∈I}be a set ofT1-spaces, and let Y be dense in an open subspace ofXI:=∏i∈IXi. Ifχ(qi,Xi)⩽ωfor everyi∈Iand every q in theGδ-closure of Y inXI, then for every regular space Z with a regularGδ-diagonal, every continuous functionf:Y→Zextends continuously over theGδ-closure of Y inXI.Some examples are cited to show that the hypothesis χ(qi,Xi)⩽ω cannot be replaced by the weaker hypothesis ψ(qi,Xi)⩽ω.

The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X, we use ↓USC F (X) and ↓C F (X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0, 1] with the subspace topologies of the hyperspace Cld F (X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology. In this paper, we shall show that there exists a homeomorphism h: ↓USC F (X) → Q = [−1, 1] ω such that h(↓C F (X)) = c 0 = {(x n ) ∈ Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.

Orthocompactness versus normality in hyperspaces

For a regular space X, 2 X denotes the collection of all non-empty closed sets of X with the Vietoris topology and K ( X ) denotes the collection of all non-empty compact sets of X with the subspace topology of 2 X . In this paper, we will prove: • K ( γ ) is orthocompact iff either cf γ ⩽ ω or γ is a regular uncountable cardinal, as a corollary normality and orthocompactness of K ( γ ) are equivalent for every non-zero ordinal γ. We present its two proofs, one proof uses the elementary submodel techniques and another does not. This also answers Question C of Kemoto (2007) [4]. Moreover we discuss the natural question whether 2 ω is orthocompact or not. We prove that • 2 ω is orthocompact iff it is countably metacompact, • the hyperspace K ( S ) of the Sorgenfrey line S is orthocompact therefore so is the Sorgenfrey plane S 2 .

Perturbative Quantum Error Correction

We derive simple necessary and sufficient conditions under which a quantum channel obtained from an arbitrary perturbation from the identity can be reversed on a given code to the lowest order in fidelity. We find the usual Knill-Laflamme conditions applied to a certain operator subspace which, for a generic perturbation, is generated by the Lindblad operators. For a weak interaction with an environment, the error space to be corrected is a subspace of that spanned by the interaction operators, selected by the environment's initial state.

Hyperinvariant subspaces for operators having a compact part

It is well known that if T = A ⊕ B , where A is compact, then T has a nontrivial hyperinvariant subspace. In this paper, we try to solve the hyperinvariant subspace problem for operators which have a compact part. Our main result is that if A is compact, then either ( A ⁎ 0 B ) or ( A 0 ⁎ B ) has a nontrivial hyperinvariant subspace.

Random maximal isotropic subspaces and Selmer groups

Under suitable hypotheses, we construct a probability measure on the set of closed maximal isotropic subspaces of a locally compact quadratic space over F p \mathbb {F}_p . A random subspace chosen with respect to this measure is discrete with probability 1 1 , and the dimension of its intersection with a fixed compact open maximal isotropic subspace is a certain nonnegative-integer-valued random variable. We then prove that the p p -Selmer group of an elliptic curve is naturally the intersection of a discrete maximal isotropic subspace with a compact open maximal isotropic subspace in a locally compact quadratic space over F p \mathbb {F}_p . By modeling the first subspace as being random, we can explain the known phenomena regarding distribution of Selmer ranks, such as the theorems of Heath-Brown, Swinnerton-Dyer, and Kane for 2 2 -Selmer groups in certain families of quadratic twists, and the average size of 2 2 - and 3 3 -Selmer groups as computed by Bhargava and Shankar. Our model is compatible with Delaunay’s heuristics for p p -torsion in Shafarevich-Tate groups, and predicts that the average rank of elliptic curves over a fixed number field is at most 1 / 2 1/2 . Many of our results generalize to abelian varieties over global fields.

Equivariant $K$-theory and the Chern character for discrete groups

Let X be a compact Hausdorff space, let Γ be a discrete group that acts continuously on X from the right, define X = {(x, γ) ∈ X×Γ : x ·γ = x}, and let Γ act on X via the formula (x, γ) · α = (x · α,α−1γα). Results of P. Baum and A. Connes, along with facts about the Chern character, imply that Ki Γ(X) and K i( X/Γ) are isomorphic up to torsion for i = 0, 1. In this paper, we present an example where the groups Ki Γ(X) and K i( X/Γ) are not isomorphic. Let Γ be a finite discrete group acting continuously on a compact Hausdorff space X from the right. Define X = {(x, γ) ∈ X × Γ : x · γ = x} and endow X with the subspace topology that it inherits from X × Γ. The group Γ acts on X via the formula (x, γ) ·α = (x ·α, αγα), and we can consider the orbit space X/Γ. Theorem 1.19 in [1] states that there exist isomorphisms K Γ(X)⊗ C ∼= ∞ ∑

Existence of Nonzero, Linear and Continuous Operator between Two Musielak-Orlicz Spaces

Let $X, Y$ be linear topological subspaces of the space of all measurable functions over a $\sigma$-finite, atomless and complete measure space $\Omega$. The question of existence of a nonzero, linear continuous operator $T: X \to Y$ is a natural extension of the question "does X admit a nonzero, linear continuous functional?". In case of $X = L^\Phi(\Omega)$, $Y = L^\Psi(\Omega)$ being Orlicz spaces Ph.Turpin ([11]) gave a criterion telling when there is no nonzero, linear and continuous operator between $L^\Phi(\Omega)$ and $L^\Psi(\Omega)$. That result was extended to the case of Musielak-Orlicz spaces by A.K.Kalindé, R.Płuciennik and M.Wisła ([2, 8]) but only necessary conditions have been presented by them - in fact conditions assuring that there is no nonzero, linear and continuous operator between $L^\Phi(\Omega)$ and $L^\Psi(\Omega)$. In this paper we give inverse theorems to the Kalindé and Płuciennik-Wisła theorems. Generally speaking we state that there exists a nonzero, linear and continuous operator if and only if it there exists a set $A \subset \Omega$ of positive and finite measure such that the inclusion operator $i: L^\Phi(A) \to L^\Psi(A)$ is nonzero and continuous.

Invariant subspaces and reducing subspaces of weighted Bergman space over polydisc

In this paper, we study the invariant subspaces and reducing subspace of the weighted Bergman space over polydisc. The minimal reducing subspaces of a class of analytic Toeplitz operators are completely described, and Beurling-type theorem of some invariant subspaces of Toeplitz operators T zi (1≤ i ≤ n )on the weighted Bergman space over polydisc is also obtained.

Period spaces for Hodge structures in equal characteristic

We develop the analog in equal positive characteristic of Fontaine’s theory for crystalline Galois representations of a p-adic eld. In particular we describe the analog of Fontaine’s functor which assigns to a crystalline Galois representation an isocrystal with a Hodge ltration. In equal characteristic the role of isocrystals and Hodge ltrations is played by z-isocrystals and Hodge-Pink structures. The latter were invented by Pink. Our rst main result in this article is the analog of the Colmez-Fontaine Theorem that \weakly implies admissible. Next we construct period spaces for Hodge-Pink structures on a xed z-isocrystal. These period spaces are analogs of the Rapoport-Zink period spaces for Fontaine’s ltered isocrystals in mixed characteristic and likewise are rigid analytic spaces. For our period spaces we prove the analog of a conjecture of Rapoport and Zink stating the existence of a \universal local system on a Berkovich open subspace of the period space. As a consequence of \weakly implies admissible this Berkovich open subspace contains every classical rigid analytic point of the period space. As the principal tool to demonstrate these results we use the analog of Kedlaya’s Slope Filtration Theorem which we also formulate and prove here.

Topological structure of fuzzy soft sets

In this paper, the concept of fuzzy soft topology is introduced and some of its structural properties such as neighborhood of a fuzzy soft set, interior fuzzy soft set, fuzzy soft basis, fuzzy soft subspace topology are studied.

Compactification-like extensions

Let $X$ be a space. A space $Y$ is called an extension of $X$ if $Y$ contains $X$ as a dense subspace. For an extension $Y$ of $X$ the subspace $Y\backslash X$ of $Y$ is called the remainder of $Y$. Two extensions of $X$ are said to be equivalent if there is a homeomorphism between them which fixes $X$ pointwise. For two (equivalence classes of) extensions $Y$ and $Y'$ of $X$ let $Y\leq Y'$ if there is a continuous mapping of $Y'$ into $Y$ which fixes $X$ pointwise. Let $P$ be a topological property. An extension $Y$ of $X$ is called a $P$-extension of $X$ if it has $P$. If $P$ is compactness then $P$-extensions are called ompactifications. The aim of this article is to introduce and study classes of extensions (which we call compactification-like $P$-extensions, where $P$ is a topological property subject to some mild requirements) which resemble the classes of compactifications of locally compact spaces. We formally define compactification-like $P$-extensions and derive some of their basic properties, and in the case when the remainders are countable, we characterize spaces having such extensions. We will then consider the classes of compactification-like $P$-extensions as partially ordered sets. This consideration leads to some interesting results which characterize compactification-like $P$-extensions of a space among all its Tychonoff $P$-extensions with compact remainder. Furthermore, we study the relations between the order-structure of classes of compactification-like $P$-extensions of a Tychonoff space $X$ and the topology of a certain subspace of its outgrowth $\beta X\backslash X$. We conclude with some applications, including an answer to an old question of S. Mr\'{o}wka and J.H. Tsai: For what pairs of topological properties $P$ and $Q$ is it true that every locally-$P$ space with $Q$ has a one-point extension with both $P$ and $Q$?

Hyperinvariant subspaces for operators having a normal part

Hyperinvariant subspaces for operators having a normal part

Operator subspaces of $\mathcal{L}(H)$ with induced matrix orderings

Operator subspaces of $\mathcal{L}(H)$ with induced matrix orderings