Identity Operator Research Articles

Closed ideals of operators on the Tsirelson and Schreier spaces

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals.Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows:•the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements;•for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, andI⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

Correction to: C*-algebras, H*-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff and abelian groups

Correction to: C*-algebras, H*-algebras and trace ideals of pseudo-differential operators on locally compact, Hausdorff and abelian groups

Electronic Dynamics through Conical Intersections via Quasiclassical Mapping Hamiltonian Methods

In this work, we investigate the ability of different quasiclassical mapping Hamiltonian methods to simulate the dynamics of electronic transitions through conical intersections. The analysis is carried out within the framework of the linear vibronic coupling (LVC) model. The methods compared are the Ehrenfest method, the symmetrical quasiclassical method, and several variations of the linearized semiclassical (LSC) method, including ones that are based on the recently introduced modified representation of the identity operator. The accuracy of the various methods is tested by comparing their predictions to quantum-mechanically exact results obtained via the multiconfiguration time-dependent Hartree (MCTDH) method. The LVC model is found to be a nontrivial benchmark model that can differentiate between different approximate methods based on their accuracy better than previously used benchmark models. In the three systems studied, two of the LSC methods are found to provide the most accurate description of electronic transitions through conical intersections.

On Square-Integrable Representations of A Lie Group of 4-Dimensional Standard Filiform Lie Algebra

In this paper, we study irreducible unitary representations of a real standard filiform Lie group with dimension equals 4 with respect to its basis. To find this representations we apply the orbit method introduced by Kirillov. The corresponding orbit of this representation is genereric orbits of dimension 2. Furthermore, we show that obtained representation of this group is square-integrable. Moreover, in such case , we shall consider its Duflo-Moore operator as multiple of scalar identity operator. In our case that scalar is equal to one.

Homotopy analysis of the Lippmann–Schwinger equation for seismic wavefield modelling in strongly scattering media

SUMMARYWe present an application of the homotopy analysis method for solving the integral equations of the Lippmann–Schwinger type, which occurs frequently in acoustic and seismic scattering theory. In this method, a series solution is created which is guaranteed to converge independent of the scattering potential. This series solution differs from the conventional Born series because it contains two auxiliary parameters ϵ and h and an operator H that can be selected freely in order to control the convergence properties of the scattering series. The ϵ-parameter which controls the degree of dissipation in the reference medium (that makes the wavefield updates localized in space) is known from the so-called convergent Born series theory; but its use in conjunction with the homotopy analysis method represents a novel feature of this work. By using H = I (where I is the identity operator) and varying the convergence control parameters h and ϵ, we obtain a family of scattering series which reduces to the conventional Born series when h = −1 and ϵ = 0. By using H = γ where γ is a particular pre-conditioner and varying the convergence control parameters h and ϵ, we obtain another family of scattering series which reduces to the so-called convergent Born series when h = −1 and ϵ ≥ ϵc where ϵc is a critical dissipation parameter depending on the largest value of the scattering potential. This means that we have developed a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. By performing a series of 12 numerical experiments with a strongly scattering medium, we illustrate the effects of varying the (ϵ, h, H)-parameters on the convergence properties of the new homotopy scattering series. By using (ϵ, h, H) = (0.5, −0.8, I) we obtain a new scattering series that converges significantly faster than the convergent Born series. The use of a non-zero dissipation parameter ϵ seems to improve on the convergence properties of any scattering series, but one can now relax on the requirement ϵ ≥ ϵc from the convergent Born series theory, provided that a suitable value of the convergence control parameter h and operator H is used.

Benchmarking Quasiclassical Mapping Hamiltonian Methods for Simulating Electronically Nonadiabatic Molecular Dynamics.

Quasi-classical mapping Hamiltonian methods have recently emerged as a promising approach for simulating electronically nonadiabatic molecular dynamics. The classical-like dynamics of the overall system within these methods makes them computationally feasible, and they can be derived based on well-defined semiclassical approximations. However, the existence of a variety of different quasi-classical mapping Hamiltonian methods necessitates a systematic comparison of their respective advantages and limitations. Such a benchmark comparison is presented in this paper. The approaches compared include the Ehrenfest method, the symmetrical quasi-classical (SQC) method, and five variations of the linearized semiclassical (LSC) method, three of which employ a modified identity operator. The comparison is based on a number of popular nonadiabatic model systems; the spin-boson model, a Frenkel biexciton model, and Tully's scattering models 1 and 2. The relative accuracy of the different methods is tested by comparing with quantum-mechanically exact results for the dynamics of the electronic populations and coherences. We find that LSC with the modified identity operator typically performs better than the Ehrenfest and standard LSC approaches. In comparison to SQC, these modified methods appear to be slightly more accurate for condensed phase problems, but for scattering models there is little distinction between them.

Davis–Wielandt shells of semi-Hilbertian space operators and its applications

In this paper we generalize the concept of Davis–Wielandt shell of operators on a Hilbert space when a semi-inner product induced by a positive operator A is considered. Moreover, we investigate the parallelism of A-bounded operators with respect to the seminorm and the numerical radius induced by A. Mainly, we characterize A-normaloid operators in terms of their A-Davis–Wielandt radii. In addition, a connection between A-seminorm-parallelism to the identity operator and an equality condition for the A-Davis–Wielandt radius is proved. This generalizes the well-known results in Chan and Chan (Oper Matrices 11(3):885–890, 2017), Zamani et al. (Linear Multilinear Algebra 67(11):2147–2158, 2019). Some other related results are also discussed.

Hermitian projections on some Banach spaces and related topics

In the setting of various complex Banach spaces we consider the questions of when a square of a Hermitian operator is also Hermitian, and when a Hermitian operator has a Hermitian square root. We also describe the structure of Hermitian projections and Hermitian square roots of the identity operator.

The Lipschitz injective hull of Lipschitz operator ideals and applications

We introduce and study the Lipschitz injective hull of Lipschitz operator ideals defined between metric spaces. We show some properties and apply the results to the ideal of Lipschitz p-nuclear operators, obtaining the ideal of Lipschitz quasi p-nuclear operators. Also, we introduce in a natural way the ideal of Lipschitz Pietsch p-integral operators and show that its Lipschitz injective hull coincide with the ideal of Lipschitz p-summing operators defined by Farmer and Johnson. Finally, we consider both ideals as Lipschitz operator ideals between a metric space and a Banach space, showing that these ideals are not of composition type. Their maximal hull and minimal kernel are also studied.

Dixmier traces and residues on weak operator ideals

We develop the theory of modulated operators in general principal ideals of compact operators. For Laplacian modulated operators we establish Connes' trace formula in its local Euclidean model and a global version thereof. It expresses Dixmier traces in terms of the vector-valued Wodzicki residue. We demonstrate the applicability of our main results in the context of log-classical pseudo-differential operators, studied by Lesch, and a class of operators naturally appearing in noncommutative geometry.

Legendre–Galerkin Methods for Third Kind VIEs and CVIEs

The main purpose of this paper is to present a spectral Legendre–Galerkin method for solving Volterra integral equations of the third kind. When the operator associated with the equivalent Volterra integral equations of second kind is compact, the resulting system produced by this spectral method is uniquely solvable and the approximate solution attains the optimal convergence order. While the related operator is noncompact, that brings a serious challenge in numerical analysis. In order to overcome this difficulty, we first decompose the original operator into three operators, one is the identity operator, the other is the contraction operator and the third one is compact. Under this decomposition, we show that the proposed method guarantees the unique solvability of the approximate equation. Moreover, we establish that the approximate solution arrives at the quasi-optimal order of global convergence. In addition, we extend this spectral method to solve the associated cordial Volterra integral equations. Finally, to confirm the theoretical results, two numerical examples are presented.

Embedding of $$BMOA_{\log }$$ into Tent Spaces and Volterra Integral Operators

In this paper, we study the boundedness and compactness of the identity operator $$I:BMOA_{\log }\rightarrow \mathcal {T}^{\infty }_{\log }(\mu )$$. As applications, we characterize the boundedness and compactness of the Volterra integral operators $$T_g$$ and $$I_g$$ on the space $$BMOA_{\log }$$. The estimations for the essential norm of $$T_g$$ and $$I_g$$ on the space $$BMOA_{\log }$$ are also given.

The lattices of families of regular sets in topological spaces

Abstract The question as to the number of sets obtainable from a given subset of a topological space using the operators derived by composing members of the set {b, i, ∨, ∧}, where b, i, ∨ and ∧ denote the closure operator, the interior operator, the binary operators corresponding to union and intersection, respectively, is called the Kuratowski {b, i, ∨, ∧}-problem. This problem has been solved independently by Sherman [21] and, Gardner and Jackson [13], where the resulting 34 plus identity operators were depicted in the Hasse diagram. In this paper we investigate the sets of fixed points of these operators. We show that there are at most 23 such families of subsets. Twelve of them are the topology, the family of all closed subsets plus, well known generalizations of open sets, plus the families of their complements. Each of the other 11 families forms a complete complemented lattice under the operations of join, meet and negation defined according to a uniform procedure. Two of them are the well known Boolean algebras formed by the regular open sets and regular closed sets, any of the others in general need not be a Boolean algebras.

Hardy's operator minus identity and power weights

Let H be the Hardy operator and I the identity operator acting on functions on the real half-line. We find optimal bounds for the operator H−I in the setting of power weights and the cases of positive decreasing functions, positive functions, and general functions. As a byproduct, we obtain some results about the optimal relations between the norms of H and its dual.

Discrete harmonic analysis associated with Jacobi expansions I: The heat semigroup

In this paper we commence the study of discrete harmonic analysis associated with Jacobi orthogonal polynomials of order (α,β). Particularly, we analyse the heat semigroup Wt(α,β), t≥0, related to the operator J(α,β)−I, where J(α,β) is the three-term recurrence relation for the normalised Jacobi polynomials and I is the identity operator. First, we prove the positivity of the operator Wt(α,β) under some suitable restrictions on the parameters α and β. In our main result, we investigate the mapping properties of the maximal operator defined by the heat semigroup in weighted ℓp-spaces using discrete vector-valued local Calderón-Zygmund theory. Moreover, we treat the Poisson semigroup by means of an appropriate subordination identity.

The Composition Operation on Spaces of Holomorphic Mappings

AbstractWe discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

Smooth Orthogonal Projections on Riemannian Manifold

We construct a decomposition of the identity operator on a Riemannian manifold M as a sum of smooth orthogonal projections subordinate to an open cover of M. This extends a decomposition on the real line by smooth orthogonal projection due to Coifman and Meyer (C. R. Acad. Sci. Paris, Sér. I Math., 312(3), 259–261 1991) and Auscher, Weiss, Wickerhauser (1992), and a similar decomposition when M is the sphere by Bownik and Dziedziul (Const. Approx., 41, 23–48 2015).

Resolvent growth condition for composition operators on the Fock space

For each analytic map psi on the complex plane mathbb {C}, we study the Ritt’s resolvent growth condition for the composition operator C_{psi} :f rightarrow fcirc psi on the Fock space {mathcal {F}}_2. We show that C_{psi} satisfies such a condition if and only if it is either compact or reduces to the identity operator. As a consequence, it is shown that the Ritt’s resolvent condition and the unconditional Ritt’s condition for C_{psi} are equivalent.

Zero point principle of ball-near identity operators and applications to implicit operator problem

Zero point principle of ball-near identity operators and applications to implicit operator problem

The Existence of Singular Traces on Simply Generated Operator Ideals

We present direct and self-contained proofs of the existence of singular traces on sufficiently large classes of operator ideals that live on the separable infinite-dimensional real Hilbert space. Our only tool is Banach’s version of the extension theorem of linear forms. Thanks to the use of dyadic representations of operators, all proofs have become straightforward.