Decomposable Operators Research Articles

Decomposable operators acting between distinct $$L^p$$-direct integrals of Banach spaces

Decomposable operators acting between distinct $$L^p$$-direct integrals of Banach spaces

Finite rank perturbations of normal operators: Spectral idempotents and decomposability

We prove that a large class of finite rank perturbations of diagonal operators and, in general, of diagonalizable normal operators of multiplicity one acting boundedly on a separable, infinite dimensional complex Hilbert space are decomposable operators in the sense of Colojoară and Foiaş [1]. Consequently, every operator T in such a class has a rich spectral structure and plenty of non-trivial closed hyperinvariant subspaces which extends, in particular, previous theorems of Foiaş, Jung, Ko and Pearcy [5–7], Fang and J. Xia [3] and the authors [8,9] on an open question posed by Pearcy in the seventies.

DOSnet as a non-black-box PDE solver: When deep learning meets operator splitting

Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by decomposable operators, we show that the classical “operator splitting” technique can be adapted to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics, and is more efficient and flexible than the classical numerical schemes and standard DNNs. To demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schrödinger equations which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.

Sufficient Conditions for Extended Spectral Decomposable Multi-Valued linear operators

Sufficient Conditions for Extended Spectral Decomposable Multi-Valued linear operators

Random sets and Choquet-type representations

As appropriate generalizations of convex combinations with uncountably many terms, we introduce the so-called Choquet combinations, Choquet decomposable combinations and Choquet convex decomposable combinations, as well as their corresponding hull operators acting on the power sets of Lebesgue-Bochner spaces. We show that Choquet hull coincides with convex hull in the finite-dimensional setting, yet Choquet hull tends to be larger in infinite dimensions. We also provide a quantitative characterization of Choquet hull, without any topological or algebraic assumptions on the underlying set. Furthermore, we show that the Choquet decomposable hull of a set coincides with its strongly closed decomposable hull and the Choquet convex decomposable hull of a set coincides with the Choquet decomposable hull of its convex hull. It turns out that the measurable selections of a closed-valued multifunction form a Choquet decomposable set and those of a closed convex-valued multifunction form a Choquet convex decomposable set. Finally, we investigate the operator-type features of Choquet decomposable and Choquet convex decomposable hull operators when applied in succession.

Necessary conditions for extended spectral decomposable multivalued linear operators

In this paper, we use subsets of the Riemann sphere and specific types of invariant linear subspaces to introduce the extended spectral decomposable multivalued linear operators (linear relations) in Banach spaces. We also introduce the extended Bishop's property, the extended relatively single-valued extension property and the extended Dunford's property. More importantly, we show that these properties are three necessary conditions for a linear relation to be extended spectral decomposable.

Local Spectral Properties Under Conjugations

In this paper, we study some local spectral properties of operators having form JTJ, where J is a conjugation on a Hilbert space H and Tin L(H). We also study the relationship between the quasi-nilpotent part of the adjoint T^* and the analytic core K(T) in the case of decomposable complex symmetric operators. In the last part we consider Weyl type theorems for triangular operator matrices for which one of the entries has form JTJ, or has form JT^*J. The theory is exemplified in some concrete cases.

Nonlinear Spectrum and Fixed Point Index for a Class of Decomposable Operators

We study a class of nonlinear operators that can be written as the composition of a linear operator and a nonlinear map. We obtain results on fixed point index based on parameters that are related to the definitions of nonlinear spectra. As a particular case, existence of positive solutions for a second-order differential equation with separated boundary conditions is proved. The result also provides a spectral interval for the corresponding Hammerstein integral operator.

On the representation of local indistinguishability operators

This paper studies local indistinguishability operators, i.e., symmetric and transitive fuzzy relations that do not need to be reflexive. This is an important generalization of global indistinguishability relations (fuzzy relations satisfying the reflexivity property in addition) because there are interesting families of fuzzy relations that are non-reflexive. One case are decomposable relations, that are generated by a fuzzy subset and contains the t-norms as an important subfamily. Also the relations associated naturally to fuzzy subgroups are local indistinguishability operators. In this paper these relations will be studied stressing the way they can be generated. A representation theorem will be proved and related to the concepts of extensionality and of fuzzy rough set. Decomposable local indistinguishability operators will also be studied and related with one-dimensional ones in the sense of the previous representation theorem. The presence of these relations in the study of fuzzy subgroups will also be analyzed.

Decomposability of Krein space operators

In this paper, we review some properties in the local spectral theory and various subclasses of decomposable operators. We prove that every Krein space selfadjoint operator having property (?) is decomposable, and clarify the relation between decomposability and property (?) for J-selfadjoint operators. We prove the equivalence of these properties for J-selfadjoint operators T and T* by using their local spectra and local spectral subspaces.

Geometry of joint spectra and decomposable operator tuples

Joint spectra of tuples of operators are subsets in complex projective space. We investigate the relationship between the geometry of the spectrum and the properties of the operators in the tuple when these operators are self-adjoint. In the case when the spectrum contains an algebraic hypersurface passing through an isolated spectral point of one of the operators we give necessary and sufficient geometric conditions for the operators in the tuple to have a common reducing subspace. We also address spectral continuity and obtain a norm estimate for the commutant of a pair of self-adjoint matrices in terms of the Hausdorff distance of their joint spectrum to a family of lines.

On Random Normal Operators and Their Spectral Measures

The main aim of this paper is to introduce and study the subclass of not necessarily continuous, normal random operators, establishing connections with other subclasses of random operators, as well as with the existing concept of random projection operator-valued measure. Hence, after recalling some basic facts regarding random operators on a complex separable Hilbert space, theorems about transforming the class of not necessarily continuous decomposable random operators into the class of purely contractive random operators are proved. These are applied to obtain integral representations for not necessarily continuous normal or self-adjoint random operators on a Hilbert space with respect to the corresponding random projection operator-valued measures.

Двойственная задача к эколого-экономической межотраслевой балансовой модели с нелинейной зависимостью и монотонно разложимым оператором

Двойственная задача к эколого-экономической межотраслевой балансовой модели с нелинейной зависимостью и монотонно разложимым оператором

Multinormed semifinite von Neumann algebras, unbounded operators and conditional expectations

The present paper is devoted to classification of multinormed W⁎-algebras in terms of their bounded parts. We obtain a precise description of the bornological predual of a multinormed W⁎-algebra, which is reduced to a multinormed noncommutative L1-space in the semifinite case. A multinormed noncommutative L2-space is obtained as the union space of a commutative domain in a semifinite von Neumann algebra. Multinormed W⁎algebras of type I are described as locally bounded decomposable (unbounded) operators associated with a measurable covering.

Pivotal decomposition schemes inducing clones of operations

We study pivotal decomposition schemes and investigate classes of pivotally decomposable operations. We provide sufficient conditions on pivotal operations that guarantee that the corresponding classes of pivotally decomposable operations are clones, and show that under certain assumptions these conditions are also necessary. In the latter case, the pivotal operation together with the constant operations generate the corresponding clone.

Geometry of the spectral semidistance in Banach algebras

Let A be a unital Banach algebra over ℂ, and suppose that the nonzero spectral values of a and b ∈ A are discrete sets which cluster at 0 ∈ ℂ, if anywhere. We develop a plane geometric formula for the spectral semidistance of a and b which depends on the two spectra, and the orthogonality relationships between the corresponding sets of Riesz projections associated with the nonzero spectral values. Extending a result of Brits and Raubenheimer, we further show that a and b are quasinilpotent equivalent if and only if all the Riesz projections, p(α, a) and p(α, b), correspond. For certain important classes of decomposable operators (compact, Riesz, etc.), the proposed formula reduces the involvement of the underlying Banach space X in the computation of the spectral semidistance, and appears to be a useful alternative to Vasilescu’s geometric formula (which requires the knowledge of the local spectra of the operators at each 0 ≠ x ∈ X). The apparent advantage gained through the use of a global spectral parameter in the formula aside, various methods of complex analysis can then be employed to deal with the spectral projections; we give examples illustrating the usefulness of the main results.

Extensions of spaces of analytic functions via pointwise limits of bounded sequences and two integral operators on generalized Bloch spaces

In [2], operators $$P_\mu f(z):=-\frac{1}{(1-z)^{\mu+1}} \int \limits_1^z f(\zeta)(1-\zeta)^{\mu} \,d\zeta$$ and $$Q_\mu f(z):=(1-z)^{\mu-1} \int\limits_0^z f(\zeta)(1-\zeta)^{-\mu} \,d \zeta\quad (z \in \mathbb{D})$$ were investigated in the setting of the analytic Besov spaces Bp, 1 ≤ p ≤ ∞, and the little Bloch space B∞,0. In particular, for X = Bp, 1 ≤ p < ∞, or X = B∞,0, the spectra, essential spectra of Pμ, and Qμ in \({\mathcal {L}(X),}\) together with one sided analytic resolvents in the Fredholm regions of Pμ, and Qμ were obtained along with an explicit strongly decomposable operator extending Qμ and simultaneously lifting Pμ. In the current paper, we extend the spectral analysis to generalized Bloch spaces using a modification of a construction due to Aleman and Persson, [3].

Common entanglement witnesses and their characteristics

We investigate the issue of finding common entanglement witness for certain class of states and extend this study to the case of Schmidt number witnesses. We also introduce the notion of common decomposable and non-decomposable witness operators which is specially useful for constructing a common witness where one of the entangled states is with a positive partial transpose. Our approach is illustrated with the help of suitable examples of qutrit systems.

Spectral decomposability of rank-one perturbations of normal operators

This paper is a continuation of the study by Foias, Jung, Ko, and Pearcy (2007) [4] and Foias, Jung, Ko, and Pearcy (2008) [5] of rank-one perturbations of diagonalizable normal operators. In Foias, Jung, Ko, and Pearcy (2007) [4] we showed that there is a large class of such operators each of which has a nontrivial hyperinvariant subspace, and in Foias, Jung, Ko, and Pearcy (2008) [5] we proved that the commutant of each of these rank-one perturbations is abelian. In this paper we show that the operators considered in Foias, Jung, Ko, and Pearcy (2007) [4] have more structure – namely, that they are decomposable operators in the sense of Colojoară and Foias (1968) [1].

Approximation and Similarity Classification of Stably Finitely Strongly Irreducible Decomposable Operators

AbstractLet 𝓗 be a complex separable Hilbert space and ℒ(𝓗) denote the collection of bounded linear operators on 𝓗. In this paper, we show that for any operator A ∈ ℒ(𝓗), there exists a stably finitely (SI) decomposable operator A∈, such that ‖A−A∈‖ 𝓗 ∈ andA′(A∈)/ rad A′(A∈) is commutative, where rad A′(A∈) is the Jacobson radical of A′(A∈). Moreover, we give a similarity classification of the stably finitely decomposable operators that generalizes the result on similarity classification of Cowen–Douglas operators given by C. L. Jiang.