Subspaces Of Operators Research Articles

Generalizations of the relation of quasisimilarity for operators

In this note we first briefly review the progress on the hyperinvariant subspace problem for operators on Hilbert space made possible by the equivalence relation of ampliation quasisimilarity recently introduced in [7].Then we introduce another equivalence relation, which we call pluquasisimilarity, with bigger equivalence classes than ampliation quasisimilarity but verydifferent in appearance, which preserves the existence of hyperinvariant sub-spaces for operators, and thus may be useful in the future. We also comparethese with two other equivalence relations, injection-similarity and completeinjection-similarity, introduced long ago by Sz.-Nagy and Foias in [13].

Selectors for dense subsets of function spaces

Let USCp⋆(X) be the topological space of real upper semicontinuous bounded functions defined on X with the subspace topology of the product topology on RX. Φ˜↑,Ψ˜↑ are the sets of all upper sequentially dense, upper dense or pointwise dense subsets of USCp⋆(X), respectively. We prove several equivalent assertions to that USCp⋆(X) satisfies the selection principles S1(Φ˜↑,Ψ˜↑), including a condition on the topological space X.We prove similar results for the topological space Cp⋆(X) of continuous bounded functions.Similar results hold true for the selection principles Sfin(Φ˜↑,Ψ˜↑).

Reducing Subspaces of Analytic Toeplitz Operators on the Bergman Space of the Annulus

Let $$\mathbb {A}_{r}$$ be the annulus $$\{z\mid \,|z|<r<1\}$$ in the complex plane, $$L_{a}^{2}(\mathbb {A}_{r})$$ be the Bergman space on $$\mathbb {A}_{r}$$ , B be a finite Blaschke product $$B(z)=e^{i\theta }\prod \nolimits _{i=1}^{N}\frac{z-\alpha _{i}}{1-\overline{\alpha _{i}}z}$$ with $$|\alpha _{i}|<r$$ for $$1\le i\le N$$ . In this case, local inverses of B on $$\mathbb {A}_{r}$$ consist of a cyclic group with order N. It is shown that there is an one-to-one correspondence between a minimal reducing subspace of the Toeplitz operator $$T_{B}$$ on $$L_{a}^{2}(\mathbb {A}_{r})$$ and a character of the cyclic group, reducing subspaces of Toeplitz operators are studied from an algebraic point of view and Douglas and Kim’s result is generalized.

A scalable estimator of sets of integral operators

The main objective of this work is to estimate a low dimensional subspace of operators in order to improve the identifiability of blind inverse problems. We propose a scalable method to find a subspace of low-rank tensors that simultaneously approximates a set of integral operators. The method can be seen as a generalization of tensor decomposition models, which was never used in this context. In addition, we propose to construct a convex subset of in order to further reduce the search space. We provide theoretical guarantees on the estimators and a few numerical results.

Homogeneous Components of a CDH Fuzzy Space

&lt;span style="font-size: medium;"&gt;&lt;span style="font-size: medium;"&gt;&lt;span style="font-size: 10px;"&gt;We prove that fuzzy homogeneous components of a CDH fuzzy topological space (X,T) are clopen and also they are CDH topological subspaces of its 0-cut topological space (X,T0).&lt;/span&gt;&lt;br /&gt;&lt;/span&gt;&lt;/span&gt;&lt;span style="font-size: medium;"&gt; &lt;/span&gt;

Instanton bundles on the flag variety F(0,1,2)

Instanton bundles on $\mathbb{P}^3$ have been at the core of the research in Algebraic Geometry during the last thirty years. Motivated by the recent extension of their definition to other Fano threefolds of Picard number one, we develop the theory of instanton bundles on the complete flag variety $F:=F(0,1,2)$ of point-lines on $\mathbb{P}^2$. After giving for them two different monadic presentations, we use it to show that the moduli space $MI_F(k)$ of instanton bundles of charge $k$ is a geometric GIT quotient and the open subspace $MI^s_F(k)\subset MI_F(k)$ of stable instanton bundles has a generically smooth component of dim $8k-3$. Finally we study their locus of jumping conics.

Some Properties of Interior and Closure in General Topology

We present the necessary and sufficient conditions that the intersection of an open set and a closed set becomes either an open set or a closed set. As their dualities, we further introduce the necessary and sufficient conditions that the union of a closed set and an open set becomes either a closed set or an open set. Moreover, we give some necessary and sufficient conditions for the validity of U ∘ ∪ V ∘ = ( U ∪ V ) ∘ and U ¯ ∩ V ¯ = U ∩ V ¯ . Finally, we introduce a necessary and sufficient condition for an open subset of a closed subspace of a topological space to be open. As its duality, we also give a necessary and sufficient condition for a closed subset of an open subspace to be closed.

A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

In this paper, we show that for Tin B({mathcal {H}}), if {mathcal {M}} is almost-invariant for T, then every maximal almost-invariant subspace of {mathcal {M}} is of codimension 1 in {mathcal {M}}, where {mathcal {H}} is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on {mathcal {H}}. Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.

Formal ball monads

The formal ball construction B is a central tool of quasi-metric space theory. We show that it induces monads on certain natural categories of quasi-metric spaces, with 1-Lipschitz maps as morphisms, or with 1-Lipschitz continuous maps as morphisms. Those are left Kock-Zöberlein monads, and that allows us to characterize their algebras exactly. As an application, we study so-called Lipschitz regular spaces, a natural class of spaces that contain all standard algebraic quasi-metric spaces with relatively compact balls, in particular all metric spaces whose closed balls are compact. There are other Lipschitz regular spaces, as we show, and notably all B-algebras. That includes all spaces of formal balls, with their d+-Scott topology. The value of Lipschitz regularity is that, for a Lipschitz regular standard quasi-metric space X,d, the space LX of lower semicontinuous maps from X to R‾+, with the Scott topology, retracts onto each of the spaces Lα(X,d) of α-Lipschitz continuous maps, and that the subspace topology on the latter coincides with the Scott topology.

Dual Zariski Topology on Comultiplication Modules

This paper deals with dual Zariski topology on comultiplication modules. We define a subspace topology of dual Zariski topology on comultiplication modules and study some properties of this subspace topology. We prove that XsN is an Artinian topological space if and only if M satisfies the SN-condition.

A characterization of quantum Markov semigroups of weak coupling limit type

We characterize generators of quantum Markov semigroups leaving invariant a maximal abelian purely atomic algebra and certain operator subspaces associated with it in a natural way. From this result, we also establish a characterization of generators of quantum Markov semigroups of weak coupling limit type associated with a nondegenerate Hamiltonian.

Definable one-dimensional topologies in O-minimal structures

We consider definable topological spaces of dimension one in o-minimal structures, and state several equivalent conditions for when such a topological space $$\left( X,\tau \right) $$ is definably homeomorphic to an affine definable space (namely, a definable subset of $$M^{n}$$ with the induced subspace topology). One of the main results says that it is sufficient for X to be regular and decompose into finitely many definably connected components.

Convex analysis and non-trivial invariant subspaces

We establish sufficient conditions for the existence of invariant subspaces for operators on real Banach spaces, and we investigate the behaviour of operators without such subspaces. All proofs are elementary.

Localizing algebras and diagonal operators

The notion of a localizing algebra was introduced by Lomonosov, Radjavi, and Troitsky as a side condition to build invariant subspaces for operators on Banach spaces. The goal of this paper is to show that the unital algebra generated by a single diagonal operator on a separable Hilbert space is localizing if and only if there is a non-zero compact operator in the weak closure of the unit ball of the algebra .

An example of a hereditarily normal topologically finite space, which is topologically infinite relative to the class of all its proper F σ-subspaces

Abstract A (Hausdorf) hereditarily normal (not perfectly normal) space X is constructed, which has the following properties: (a) there exists a proper open subspace of X which is homeomorphic to the whole X (i.e., the space X is topologically infinite); (b) the space is homeomorphic to none of its proper F σ {F_{\sigma}} -subspaces (i.e., the space X is topologically finite relative to the class of all its proper F σ {F_{\sigma}} -subspaces).

Some classes of Banach spaces and complemented subspaces of operators

‎The concept of $p$-$L$-limited sets and Banach spaces with the $p$-$L$-limited property ($p\in [1, \infty)$) are studied‎. ‎Some characterizations of limited $p$-convergent operators are obtained‎. ‎The complementability of some spaces of operators in the space of limited $p$-convergent operators is also investigated‎.

Separation axioms of bipolar soft topological space

We extend the study of bipolar soft topological space by defining bipolar soft separation axioms (Ti, i = 0, 1, 2, 3, 4) followed by some related theorems. The concept of bipolar soft subspace topology is also discussed, followed by examples to illustrate these concepts.

Extended Contact Algebras and Internal Connectedness

The notion of contact algebra is one of the main tools in the region-based theory of space. It is an extension of Boolean algebra with an additional relation C, called contact. Standard models of contact algebras are topological and are the contact algebras of regular closed sets in a given topological space. In such a contact algebra we add the predicate of internal connectedness with the following meaning—a regular closed set is internally connected if and only if its interior is a connected topological space in the subspace topology. We add also a ternary relation $$\vdash $$ meaning that the intersection of the first two arguments is included in the third. In this paper the extension of a Boolean algebra with $$\vdash $$ , contact and internal connectedness, satisfying certain axioms, is called an extended contact algebra. We prove a representation theorem for extended contact algebras and thus obtain an axiomatization of the theory, consisting of the universal formulas, true in all topological contact algebras with added relations of internal connectedness and $$\vdash $$ .

Joint Reducing Subspaces of Multiplication Operators and Weight of Multi-variable Bergman Spaces

This paper mainly concerns a tuple of multiplication operators defined on the weighted and unweighted multi-variable Bergman spaces, their joint reducing subspaces and the von Neumann algebra generated by the orthogonal projections onto these subspaces. It is found that the weights play an important role in the structures of lattices of joint reducing subspaces and of associated von Neumann algebras. Also, a class of special weights is taken into account. Under a mild condition it is proved that if those multiplication operators are defined by the same symbols, then the corresponding von Neumann algebras are *-isomorphic to the one defined on the unweighted Bergman space.

Homogeneity and rigidity in Erdös spaces

We investigate the homogeneity of topological subspaces of separable Hilbert space, akin to the spaces with all points rational or all points irrational, so-called Erd\H{o}s spaces. We provide a non-homogeneous example, that is based on one set of coordinates using, and a rigid example, based on a sequence of coordinate sets.