Subspaces Of Operators Research Articles

Riccati equations and Toeplitz-Berezin type symbols on Dirichlet space of unit ball

The present paper mainly gives some applications of Berezin type symbols on the Dirichlet space of unit ball. We study the solvability of some Riccati operator equations of the form XAX + XB - CX = D related to harmonic Toeplitz operators on the Dirichlet space. Especially, the invariant subspaces of Toeplitz operators are also considered.

On Nano Resolvable Spaces

This paper introduces nano resolvable spaces and nano irresolvable spaces. Also, a new form of nano subspace topology is established. Several new characterizations of nano strongly irresolvable spaces are found and precise relationships are noted between nano strongly irresolvability and nano irresolvable space. Some weaker forms related to nano irresolvable are discussed. Also, comparisons between them are given.

Introduction to neutrosophic soft topological space

The primary aim of this paper is to construct a topology on a neutrosophic soft set (NSS). The notion of neutrosophic soft interior, neutrosophic soft closure, neutrosophic soft neighbourhood, neutrosophic soft boundary, regular NSS are introduced and some of their basic properties are studied in this paper. Then the base for neutrosophic soft topology and subspace topology on NSS have been defined with suitable examples. Some related properties have been developed, too. Moreover, the concept of separation axioms on neutrosophic soft topological space have been introduced along with investigation of several structural characteristics.

Reducing subspaces for multiplication operators on the Dirichlet space through local inverses and Riemann surfaces

Abstract This paper gives a full characterization of the reducing subspaces for the multiplication operator Mϕ on the Dirichlet space with symbol of finite Blaschke product ϕ of order 5I 6I 7. The reducing subspaces of Mϕ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surfaces to study the reducing subspaces of Mϕ on the Bergman space. By this means, we determine the reducing subspaces of Mϕ on the Dirichlet space and answer some questions of Douglas-Putinar-Wang in [6].

Weakly almost periodic functionals on certain Banach algebras

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common fixed point property.

ALGEBRAIC SPECTRAL SUBSPACES OF OPERATORS WITH FINITE ASCENT

Algebraic spectral subspaces were introduced by Johnson and Sinclair via a transnite sequence of spaces. Laursen simplified the definition of algebraic spectral subspace. Algebraic spectral subspaces are useful in automatic continuity theory of intertwining linear operators on Banach spaces. In this paper, we characterize algebraic spectral subspaces of operators with finite ascent. From this characterization we show that if T is a generalized scalar operator, then T has finite ascent.

Connectedness modulo an ideal

For a topological space X and an ideal H of subsets of X we introduce the notion of connectedness modulo H. This notion of connectedness naturally generalizes the notion of connectedness in its usual sense. In the case when X is completely regular, we introduce a subspace γHX of the Stone–Čech compactification βX of X, such that connectedness modulo H is equivalent to connectedness of βX∖γHX. In particular, we prove that when H is the ideal generated by the collection of all open subspaces of X with pseudocompact closure, then X is connected modulo H if and only if clβX(βX∖υX) is connected, and when X is normal and H is the ideal generated by the collection of all closed realcompact subspaces of X, then X is connected modulo H if and only if clβX(υX∖X) is connected. Here υX is the Hewitt realcompactification of X.

Families of $p$-adic Galois representations and $(\phi, \Gamma)$-modules

We investigate the relation between p -adic Galois representations and overconvergent (\phi,\Gamma) -modules in families. Especially we construct a natural open subspace of a family of (\phi,\Gamma) -modules, over which it is induced by a family of Galois-representations.

Convexity of Decentralized Controller Synthesis

In decentralized control problems, a standard approach is to specify the set of allowable decentralized controllers as a closed subspace of linear operators. This then induces a corresponding set of Youla–Kucera parameters. Previous work has shown that quadratic invariance of the controller set implies that the set of Youla–Kucera parameters is convex. In this technical note, we prove the converse. We thereby show that the only decentralized control problems for which the set of Youla–Kucera parameters is convex are those which are quadratically invariant. We further show that under additional assumptions, quadratic invariance is necessary and sufficient for the set of achievable closed-loop maps to be convex. We give two versions of our results. The first applies to bounded linear operators on a Banach space and the second applies to (possibly unstable) causal LTI systems in discrete or continuous time.

Contraction of Hamiltonian $K$-Spaces

In the spirit of recent work of Harada-Kaveh and Nishinou-Nohara-Ueda, we study the symplectic geometry of Popov's horospherical degenerations of complex algebraic varieties with the action of a complex linearly reductive group. We formulate an intrinsic symplectic contraction of a Hamiltonian space, which is a surjective, continuous map onto a new Hamiltonian space that is a symplectomorphism on an explicitly defined dense open subspace. This map is given by a precise formula, using techniques from the theory of symplectic reduction and symplectic implosion. We then show, using the Vinberg monoid, that the gradient-Hamiltonian flow for a horospherical degeneration of an algebraic variety gives rise to this contraction from a general fiber to the special fiber. We apply this construction to branching problems in representation theory, and finally we show how the Gel'fand-Tsetlin integrable system can be understood to arise this way.

On the topology of projective subspaces in complex Fermat varieties

Let $X$ be the complex Fermat variety of dimension $n=2d$ and degree $m > 2$. We investigate the submodule of the middle homology group of $X$ with integer coefficients generated by the classes of standard $d$-dimensional subspaces contained in $X$, and give an algebraic (or rather combinatorial) criterion for the primitivity of this submodule.

Fine properties and a notion of quasicontinuity for BV functions on metric spaces

On a metric space equipped with a doubling measure supporting a Poincaré inequality, we show that given a BV function, discarding a set of small 1-capacity makes the function continuous outside its jump set and “one-sidedly” continuous in its jump set. We show that such a property implies, in particular, that the measure theoretic boundary of a set of finite perimeter separates the measure theoretic interior of the set from its measure theoretic exterior, both in the sense of the subspace topology outside sets of small 1-capacity, and in the sense of 1-almost every curve.

Quasiaffinity and invariant subspaces

Special classes of intertwining transformations between Hilbert spaces are introduced and investigated, whose purposes are to provide partial answers to some classical questions on the existence of nontrivial invariant subspaces for operators acting on separable Hilbert spaces. The main result ensures that if an operator is \({{\mathcal D}}\)-intertwined to a normal operator, then it has a nontrivial invariant subspace.

Perturbations of invariant subspaces of compact operators

Perturbations of invariant subspaces of compact operators

On the weak limit of compact operators on the reproducing kernel Hilbert space and related questions

AbstractBy applying the so-called Berezin symbols method we prove a Gohberg- Krein type theorem on the weak limit of compact operators on the non- standard reproducing kernel Hilbert space which essentially improves the similar results of Karaev [5]: We also in terms of reproducing kernels and Berezin symbols investigate the structure of invariant subspaces of compact operators.

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $${\phi}$$ on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of $${M_\phi}$$ are orthogonal. When the order n of $${\phi}$$ is 2 or 3, we show that $${M_\phi}$$ is reducible on D if and only if $${\phi}$$ is equivalent to $${z^n}$$ . When the order of $${\phi}$$ is 4, we determine the reducing subspaces for $${M_\phi}$$ , and we see that in this case $${M_\phi}$$ can be reducible on D when $${\phi}$$ is not equivalent to $${z^4}$$ . The same phenomenon happens when the order n of $${\phi}$$ is not a prime number. Furthermore, we show that $${M_\phi}$$ is unitarily equivalent to $${M_{z^n} (n > 1)}$$ on D if and only if $${\phi = az^n}$$ for some unimodular constant a.

An invariant subspace problem for multilinear operators on finite dimensional spaces

We introduce the notion of invariant subspaces for multilinear operators from which the invariant subspace problems for multilinear and polynomial operators arise. We prove that polynomial operators acting in a finite dimensional complex space and even polynomial operators acting in a finite dimensional real space have eigenvalues. These results enable us to prove that polynomial and multilinear operators acting in a finite dimensional complex space, even polynomial and even multilinear operators acting in a finite dimensional real space have nontrivial invariant subspaces. Furthermore, we prove that odd polynomial operators acting in a finite dimensional real space either have eigenvalues or are homotopic to scalar operators; we then use this result to prove that odd polynomial and odd multilinear operators acting in a finite dimensional real space may or may not have invariant subspaces.

The open XXZ spin chain model and the topological basis realization

In this paper, it is shown that the Hamiltonian of the open spin-1 XXZ chain model can be constructed from the generators of the Birman–Murakami–Wenzl (B–M–W) algebra. Without the topological parameter d (describing the unknotted loop [Formula: see text] in topology) reducing to a fixed value, the topological basis states can be connected with the open XXZ spin chain. Then some particular properties of the topological basis states in this system have been investigated. We find that the topological basis states are the three eigenstates of a four-spin-1 XXZ chain model without boundary term. Specifically, all the spin single states of the system fall on the topological basis subspace. And the number of the spin single states of the system is equal to that of the topological basis states.

Existence of the Category DTC 2 (K) Equivalent to the Given Category KAC 2

For a given category KAC 2 , the present paper deals with the existence problem for the category DTC 2 (k), which is equivalent to KAC 2 , where DTC 2 (k) is the category whose objects are simple closed K-curves with even number l of elements in Z n , l ≠ 6, and morphisms are (digitally) K-continuous maps, and KAC 2 is a category whose objects are simple closed A-curves and morphisms are A-maps. To address this issue, the paper starts from the category denoted by KAC 1 whose objects are connected nD Khalimsky topological subspaces with Khalimsky adjacency and morphisms are A-maps in [S. E. Han and A. Sostak, Comput. Appl. Math., 32, 521–536 (2013)]. Based on this approach, in KAC 1 the paper proposes the notions of A-homotopy and A-homotopy equivalence and classifies the spaces in KAC 1 or KAC 2 in terms of the A-homotopy equivalence. Finally, the paper proves that, for Sa given category KAC 2 , there is DTC 2 (k), which is equivalent to KAC 2 .

Reducing subspaces of tensor products of weighted shifts

A unilateral weighted shift A is said to be simple if its weight sequence {αn} satisfies ∇3(αn2) ≠ 0 for all n ≥ 2. We prove that if A and B are two simple unilateral weighted shifts, then A⊗I + I ⊗B is reducible if and only if A and B are unitarily equivalent. We also study the reducing subspaces of Ak ⊗ I + I ⊗ Bl and give some examples. As an application, we study the reducing subspaces of multiplication operators \(M_{z^k + \alpha w^l }\) on function spaces.