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16 pages, 349 KiB

Open AccessArticle

Non-Linear Inner Structure of Topological Vector Spaces

by Francisco Javier García-Pacheco, Soledad Moreno-Pulido, Enrique Naranjo-Guerra and Alberto Sánchez-Alzola

Mathematics 2021, 9(5), 466; https://doi.org/10.3390/math9050466 - 25 Feb 2021

Cited by 9 | Viewed by 1721

Abstract

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the [...] Read more.

Inner structure appeared in the literature of topological vector spaces as a tool to characterize the extremal structure of convex sets. For instance, in recent years, inner structure has been used to provide a solution to The Faceless Problem and to characterize the finest locally convex vector topology on a real vector space. This manuscript goes one step further by settling the bases for studying the inner structure of non-convex sets. In first place, we observe that the well behaviour of the extremal structure of convex sets with respect to the inner structure does not transport to non-convex sets in the following sense: it has been already proved that if a face of a convex set intersects the inner points, then the face is the whole convex set; however, in the non-convex setting, we find an example of a non-convex set with a proper extremal subset that intersects the inner points. On the opposite, we prove that if a extremal subset of a non-necessarily convex set intersects the affine internal points, then the extremal subset coincides with the whole set. On the other hand, it was proved in the inner structure literature that isomorphisms of vector spaces and translations preserve the sets of inner points and outer points. In this manuscript, we show that in general, affine maps and convex maps do not preserve inner points. Finally, by making use of the inner structure, we find a simple proof of the fact that a convex and absorbing set is a neighborhood of 0 in the finest locally convex vector topology. In fact, we show that in a convex set with internal points, the subset of its inner points coincides with the subset of its internal points, which also coincides with its interior with respect to the finest locally convex vector topology. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

18 pages, 377 KiB

Open AccessArticle

First Integrals of Differential Operators from SL(2,ℝ) Symmetries

by Paola Morando, Concepción Muriel and Adrián Ruiz

Mathematics 2020, 8(12), 2167; https://doi.org/10.3390/math8122167 - 4 Dec 2020

Cited by 2 | Viewed by 1535

Abstract

The construction of first integrals for

S L (2, R)

-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra

sl (2, R) .

Firstly, we provide for [...] Read more.

The construction of first integrals for

S L (2, R)

-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra

sl (2, R) .

Firstly, we provide for

n = 2

an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of

sl (2, R),

and without any kind of integration. Moreover, although there are cases when the two first integrals are functionally independent, it is proved that a second functionally independent first integral arises by a single quadrature. This result is extended for

n > 2,

provided that a solvable structure for an integrable distribution generated by the differential operator associated to the equation and one of the prolonged symmetry generators of

sl (2, R)

is known. Several examples illustrate the procedures. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

11 pages, 925 KiB

Open AccessArticle

Quasi-Probability Husimi-Distribution Information and Squeezing in a Qubit System Interacting with a Two-Mode Parametric Amplifier Cavity

by Eied. M. Khalil, Abdel-Baset. A. Mohamed, Abdel-Shafy F. Obada and Hichem Eleuch

Mathematics 2020, 8(10), 1830; https://doi.org/10.3390/math8101830 - 19 Oct 2020

Cited by 8 | Viewed by 2926

Abstract

Squeezing and phase space coherence are investigated for a bimodal cavity accommodating a two-level atom. The two modes of the cavity are initially in the Barut–Girardello coherent states. This system is studied with the SU(1,1)-algebraic model. Quantum effects are analyzed with the Husimi [...] Read more.

Squeezing and phase space coherence are investigated for a bimodal cavity accommodating a two-level atom. The two modes of the cavity are initially in the Barut–Girardello coherent states. This system is studied with the SU(1,1)-algebraic model. Quantum effects are analyzed with the Husimi function under the effect of the intrinsic decoherence. Squeezing, quantum mixedness, and the phase information, which are affected by the system parameters, exalt a richer structure dynamic in the presence of the intrinsic decoherence. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

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11 pages, 280 KiB

Open AccessArticle

Schur Lemma and Uniform Convergence of Series through Convergence Methods

by Fernando León-Saavedra, María del Pilar Romero de la Rosa and Antonio Sala

Mathematics 2020, 8(10), 1744; https://doi.org/10.3390/math8101744 - 11 Oct 2020

Cited by 2 | Viewed by 1673

Abstract

In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish [...] Read more.

In this note, we prove a Schur-type lemma for bounded multiplier series. This result allows us to obtain a unified vision of several previous results, focusing on the underlying structure and the properties that a summability method must satisfy in order to establish a result of Schur’s lemma type. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

15 pages, 310 KiB

Open AccessArticle

The N-Dimensional Uncertainty Principle for the Free Metaplectic Transformation

by Rui Jing, Bei Liu, Rui Li and Rui Liu

Mathematics 2020, 8(10), 1685; https://doi.org/10.3390/math8101685 - 1 Oct 2020

Cited by 19 | Viewed by 1959

Abstract

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some [...] Read more.

The free metaplectic transformation is an N-dimensional linear canonical transformation. This transformation operator is useful, especially for signal processing applications. In this paper, in order to characterize simultaneously local analysis of a function (or signal) and its free metaplectic transformation, we extend some different uncertainty principles (UP) from quantum mechanics including Classical Heisenberg’s uncertainty principle, Nazarov’s UP, Donoho and Stark’s UP, Hardy’s UP, Beurling’s UP, Logarithmic UP, and Entropic UP, which have already been well studied in the Fourier transform domain. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

17 pages, 336 KiB

Open AccessArticle

Regularity in Topological Modules

by Francisco Javier Garcia-Pacheco

Mathematics 2020, 8(9), 1580; https://doi.org/10.3390/math8091580 - 13 Sep 2020

Cited by 8 | Viewed by 1756

Abstract

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules [...] Read more.

The framework of Functional Analysis is the theory of topological vector spaces over the real or complex field. The natural generalization of these objects are the topological modules over topological rings. Weakening the classical Functional Analysis results towards the scope of topological modules is a relatively new trend that has enriched the literature of Functional Analysis with deeper classical results as well as with pathological phenomena. Following this trend, it has been recently proved that every real or complex Hausdorff locally convex topological vector space with dimension greater than or equal to 2 has a balanced and absorbing subset with empty interior. Here we propose an extension of this result to topological modules over topological rings. A sufficient condition is provided to accomplish this extension. This sufficient condition is a new property in topological module theory called strong open property. On the other hand, topological regularity of closed balls and open balls in real or complex normed spaces is a trivial fact. Sufficient conditions, related to the strong open property, are provided on seminormed modules over an absolutely semivalued ring for closed balls to be regular closed and open balls to be regular open. These sufficient conditions are in fact characterizations when the seminormed module is the absolutely semivalued ring. These characterizations allow the provision of more examples of closed-unit neighborhoods of zero. Consequently, the closed-unit ball of any unital real Banach algebra is proved to be a closed-unit zero-neighborhood. We finally transport all these results to topological modules over topological rings to obtain nontrivial regular closed and regular open neighborhoods of zero. In particular, if M is a topological R-module and

m^{*} \in M^{*}

is a continuous linear functional on M which is open as a map between topological spaces, then

{(m^{*})}^{- 1} (int (B))

is regular open and

{(m^{*})}^{- 1} (B)

is regular closed, for B any closed-unit zero-neighborhood in R. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

18 pages, 829 KiB

Open AccessArticle

Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

by Lvlin Luo

Mathematics 2020, 8(9), 1544; https://doi.org/10.3390/math8091544 - 9 Sep 2020

Cited by 1 | Viewed by 1931

Abstract

Let

T : H \to H

be a bounded linear operator on a separable Hilbert space

H

. In this paper, we construct an isomorphism [...] Read more.

Let

T : H \to H

be a bounded linear operator on a separable Hilbert space

H

. In this paper, we construct an isomorphism

F_{x x^{*}} : L^{2} (σ (| T - a |), μ_{| T - a |, ξ}) \to L^{2} {(σ (| (T - a)}^{*} |), μ_{{| (T - a)}^{*} |, F_{x x *}^{H} ξ})

such that

{(F_{x x^{*}})}^{2} = i d e n t i t y

and

F_{x x *}^{H}

is a unitary operator on

H

associated with

F_{x x^{*}}

. With this construction, we obtain a noncommutative functional calculus for the operator T and

F_{x x^{*}} = i d e n t i t y

is the special case for normal operators, such that

S = R_{| (S - a) |, ξ} (M_{z ϕ (z)} + a) R_{| S - a |, ξ}^{- 1}

is the noncommutative functional calculus of a normal operator S, where

a \in ρ (T)

,

R_{| T - a |, ξ} : L^{2} (σ (| T - a |), μ_{| T - a |, ξ}) \to H

is an isomorphism and

M_{z ϕ (z)} + a

is a multiplication operator on

L^{2} (σ (| S - a |), μ_{| S - a |, ξ})

. Moreover, by

F_{x x *}

we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class

B_{L e b} (H) \subset B (H)

such that T is Li-Yorke chaotic if and only if

T^{* - 1}

is for a Lebesgue operator T. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

12 pages, 265 KiB

Open AccessFeature PaperArticle

The Singular Value Expansion for Arbitrary Bounded Linear Operators

by Daniel K. Crane and Mark S. Gockenbach

Mathematics 2020, 8(8), 1346; https://doi.org/10.3390/math8081346 - 12 Aug 2020

Cited by 3 | Viewed by 4214

Abstract

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. [...] Read more.

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

10 pages, 249 KiB

Open AccessArticle

Characterizations of a Banach Space through the Strong Lacunary and the Lacunary Statistical Summabilities

by Soledad Moreno-Pulido, Giuseppina Barbieri, Fernando León-Saavedra, Francisco Javier Pérez-Fernández and Antonio Sala-Pérez

Mathematics 2020, 8(7), 1066; https://doi.org/10.3390/math8071066 - 2 Jul 2020

Cited by 2 | Viewed by 1874

Abstract

In this manuscript we characterize the completeness of a normed space through the strong lacunary (

N_{θ}

) and lacunary statistical convergence (

S_{θ}

) of series. A new characterization of weakly unconditionally Cauchy series through

N_{θ}

and [...] Read more.

In this manuscript we characterize the completeness of a normed space through the strong lacunary (

N_{θ}

) and lacunary statistical convergence (

S_{θ}

) of series. A new characterization of weakly unconditionally Cauchy series through

N_{θ}

and

S_{θ}

is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesàro convergence summability space. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

Review

Jump to: Research

14 pages, 287 KiB

Open AccessReview

History, Developments and Open Problems on Approximation Properties

by Ju Myung Kim and Bentuo Zheng

Mathematics 2020, 8(7), 1117; https://doi.org/10.3390/math8071117 - 7 Jul 2020

Viewed by 2227

Abstract

In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p-approximation property. Most recent progress on E-approximation [...] Read more.

In this paper, we give a comprehensive review of the classical approximation property. Then, we present some important results on modern variants, such as the weak bounded approximation property, the strong approximation property and p-approximation property. Most recent progress on E-approximation property and open problems are discussed at the end. Full article

(This article belongs to the Special Issue Functional Analysis, Topology and Quantum Mechanics)

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