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

The construction of first integrals for $SL(2,\mathbb{R})$-invariant nth-order ordinary differential equations is a non-trivial problem due to the nonsolvability of the underlying symmetry algebra $\mathfrak{sl}(2,\mathbb{R}).$ Firstly, we provide for $n=2$ an explicit expression for two non-constant first integrals through algebraic operations involving the symmetry generators of $\mathfrak{sl}(2,\mathbb{R}),$ 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 $\mathfrak{sl}(2,\mathbb{R})$ is known. Several examples illustrate the procedures. Full article

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

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

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

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 $m^{\ast }\in M^{\ast }$ is a continuous linear functional on M which is open as a map between topological spaces, then ${\left(m^{\ast }\right)}^{-1}\left(\mathrm{int}\left(B\right)\right)$ is regular open and ${\left(m^{\ast }\right)}^{-1}\left(B\right)$ is regular closed, for B any closed-unit zero-neighborhood in R. Full article

Let $T:\mathbb{H}\to \mathbb{H}$ be a bounded linear operator on a separable Hilbert space $\mathbb{H}$. In this paper, we construct an isomorphism $F_{x{x}^{*}}:{\mathcal{L}}^2\left(\sigma \right(|T-a\left|\right),{\mu }_{|T-a|,\xi })\to {\mathcal{L}}^2{\left(\sigma \right(|(T-a)}^{*}\left|\right),{\mu }_{{|(T-a)}^{*}|,F_{xx*}^{\mathbb{H}}\xi })$ such that ${\left(F_{x{x}^{*}}\right)}^2=identity$ and $F_{xx*}^{\mathbb{H}}$ is a unitary operator on $\mathbb{H}$ associated with $F_{x{x}^{*}}$. With this construction, we obtain a noncommutative functional calculus for the operator T and $F_{x{x}^{*}}=identity$ is the special case for normal operators, such that $S=R_{\left|\right(S-a\left)\right|,\xi }(M_{z\varphi \left(z\right)}+a)R_{|S-a|,\xi }^{-1}$ is the noncommutative functional calculus of a normal operator S, where $a\in \rho \left(T\right)$, $R_{|T-a|,\xi }:{\mathcal{L}}^2\left(\sigma \right(|T-a\left|\right),{\mu }_{|T-a|,\xi })\to \mathbb{H}$ is an isomorphism and $M_{z\varphi \left(z\right)}+a$ is a multiplication operator on ${\mathcal{L}}^2\left(\sigma \right(|S-a\left|\right),{\mu }_{|S-a|,\xi })$. Moreover, by $F_{xx*}$ we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class ${\mathcal{B}}_{Leb}\left(\mathbb{H}\right)\subset \mathcal{B}\left(\mathbb{H}\right)$ such that T is Li-Yorke chaotic if and only if $T^{*-1}$ is for a Lebesgue operator T. Full article

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

In this manuscript we characterize the completeness of a normed space through the strong lacunary ( $N_{\theta }$ ) and lacunary statistical convergence ( $S_{\theta }$ ) of series. A new characterization of weakly unconditionally Cauchy series through $N_{\theta }$ and $S_{\theta }$ is obtained. We also relate the summability spaces associated with these summabilities with the strong p-Cesàro convergence summability space. Full article

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