Abstract In this work, we investigate the existence of a mild solution and the approximate controllability of non-instantaneous impulsive stochastic integrodifferential equations driven by the Rosenblatt process in Hilbert space with the Hurst parameter H ∈ (1/2, 1). We achieve the result using the semigroup theory of bounded linear operators, Grimmer’s resolvent operator theory, and stochastic analysis. Using Krasnoselskii’s and Schauder’s fixed point theorems, we demonstrate the existence of mild solutions and the approximate controllability of the system. Finally, an example shows the potential for significant results.

Resumen En este trabajo investigamos la existencia de una solución mild y la controlabilidad aproximada de ecuaciones integro-diferenciales estocásticas no-instantáneas impulsivas dirigidas por el proceso de Rosenblatt en espacios de Hilbert con el parámetro de Hurst H ∈ (1/2, 1). Logramos este resultado usando la teoría de semigrupos de operadores lineales acotados, la teoría del operador resolvente de Grimmer y análisis estocástico. Usando los teoremas de punto fijo de Krasnoselskii y Schauder, demostramos la existencia de soluciones mild y la controlabilidad aproximada del sistema. Finalmente, un ejemplo muestra el potencial para resultados significativos.

Abstract Suppose 𝔅(H) is the Banach algebra of all bounded linear operators on a Hilbert space H with dim(H) ≥ 3. Let γ(.) denote the reduced minimum modulus of an operator. We charaterize surjective maps φ on 𝔅 (H) satisfying γ(φ(T )φ(S)) = γ(T S) (T, S ∈ 𝔅 (H)). Also, we give the general form of surjective maps on 𝔅(H) preserving the reduced minimum modulus of Jordan triple products of operators.

Resumen Suponga que 𝔅(H) es el álgebra de Banach de todos los operadores lineales acotados en un espacio de Hilbert H con dim(H) ≥ 3. Denote por γ(.) el módulo mínimo reducido de un operador. Caracterizamos las aplicaciones sobreyectivas φ en 𝔅(H) que satisfacen γ(φ(T )φ(S)) = γ(T S) (T, S ∈ 𝔅 (H)). También entregamos la forma general de las aplicaciones sobreyectivas en 𝔅(H) que preservan el módulo mínimo reducido de productos triples de Jordan de operadores. 2020 AMS Mathematics Subject Classification: 47A10, 47A13, 47A25, 47B02, 47B49.

Resumen: En este artículo, construimos una sucesión similar a la de Newton para acercarnos al cero de una función diferenciable en el sentido Fréchet no lineal definida en un espacio de Hilbert. Esta nueva sucesión utiliza el concepto del adjunto del operador, que hace que el proceso iterativo sea más manejable en la práctica en comparación al desarrollado por Kantorovich que requiere el cálculo del operador inverso en cada iteración. Dado que el cálculo del operador adjunto es más fácil en comparación con el cálculo del operador inverso que en la práctica equivale a resolver un sistema de ecuaciones, nuestra nuevo método hace que el cálculo del término de nuestra nueva sucesión sea más fácil y conveniente para la aproximación numérica. Proporcionamos un teorema de convergencia a priori de esta sucesión, donde usamos unas hipótesis equivalentes a las construidas por Kantorovich, y mostramos que nuestra nueva sucesión iterativa converge hacia la solución.

Abstract: In this paper, we build a Newton-like sequence to approach the zero of a nonlinear Fréchet differentiable function defined in Hilbert space. This new iterative sequence uses the concept of the adjoint operator, which makes it more manageable in practice compared to the one developed by Kantorovich which requires the calculation of the inverse operator in each iteration. Because the calculation of the adjoint operator is easier compared to the calculation of the inverse operator which requires in practice solving a system of linear equations, our new method makes the calculation of the term of our new sequence easier and more convenient for numerical approximations. We provide an a priori convergence theorem of this sequence, where we use hypotheses equivalent to those constructed by Kantorovich, and we show that our new iterative sequence converges towards the solution.

Abstract Assuming neutrinos to be of the Dirac type, the little group generators for the one-particle states, created off the vacuum by the field operator, are obtained, both in terms of the one-particle states themselves and in terms of creation/annihilation operators. It is shown that these generators act also as rotation operators in the Hilbert space of the states, providing three types of transformations: a helicity flip, the standard charge conjugation, and a combination of the two, up to phases. The transformations’ properties are provided in detail and their physical implications discussed. It is also shown that one of the transformations continues to hold for chiral fields without mixing them. It is argued that these results provide support for the Majorana nature of massive neutrinos.

Abstract In this paper, for a complex Hilbert space ℋ with dim ℋ ≥ 2, we study the linear maps on ℬ(ℋ), the bounded linear operators on ℋ, that preserves projections and idempotents. As a result, we characterize the linear maps on ℬ(ℋ) that preserves involutions in both directions.

Abstract In this paper the controllability of a class of impulsive neutral stochastic integro-differential systems driven by fractional Brownian motion and Poisson process in a separable Hilbert space with infinite delay is studied. The controllability result is obtained by using stochas- tic analysis and a fixed-point strategy. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

Abstract: Let ℬ(ℋ) be the algebra of all bounded linear operators on an infinite dimensional Hilbert space ℋ. In this paper, we identify the form of the unital surjective additive map φ : ℬ(ℋ)→ ℬ(ℋ)which preserves the square zero of η-Lie product of operators for some scalar η with η ≠ 0, 1, −1.

In 1952, Léon van Hove published an article, in French, with the title Les difficultés de divergences pour um modèle particulier de champ quantifié. The article is frequently cited in relation to Haag's theorem and to the issue of the existence of unitarily inequivalent representations of the canonical commutation relations in Quantum Field Theory. Summarizing in brief, it suggests a link between the appearance of divergences in perturbative Quantum Field Theory and the fact that quantum states belonging to an interacting theory do not belong to the same Hilbert space of the free theory. It also suggests that renormalization fails to provide an accurate description of the time evolution of the quantum field, although it correctly accounts for the S matrix. Due to its relevance, and to the difficulty of finding an English translation, the ideas contained in this article are proposed again here, expanded with derivations and accompanied by a discussion aimed at putting the analysis into context. We highlight the main points from the perspective of a contemporary reader, and underline the differences with the standard approach usually taught in curricular courses in Quantum Field Theory.

Resumo O método de integrais de trajetória, desenvolvido por Feynman no artigo “Space-Time Approach to Non-Relativistic Quantum Mechanics” de 1948, é uma das formulações da teoria quântica que se junta, considerando a época do artigo, a duas anteriores: (i) a formulação, de certa forma o padrão apresentado em livros-textos, desenvolvida na década de 1920 por Schrödinger, Heisenberg, Dirac, Von Neumann, Born, Jordan e outros, e que se baseia no espaço de Hilbert e operadores que atuam nesse espaço, e (ii) a descrição no espaço de fase, também conhecida como a quantização de Moyal, que se baseia na função quasi-distribuição de Wigner proposta em 1932 e na lei de Correspondência de Weyl de 1927, e conduz à estrutura matemática não comutativa baseada no produto estrela (*). Neste trabalho, apresentamos uma revisão pedagógica de pontos considerados básicos para o desenvolvimento de Feynman. Destacamos a linha de pesquisa de Dirac na busca de analogias entre a Mecânica Clássica e a Mecânica Quântica, bem como os postulados formulados por Feynman no artigo acima citado, aspectos importantes para a compreensão da teoria mas pouco conhecidos e pouco divulgados de forma completa. Um exemplo é apresentado para elucidar o método e sua relação com a equação de Schrödinger dependente do tempo.

Abstract The method of path integrals developed by Feynman in the paper “Space-Time Approach to Non-Relativistic Quantum Mechanics” of 1948, is one way of formulating the quantum theory. This development joins previous ones: (i) the formulation developed by Schrödinger, Heisenberg, Dirac, Von Neumann, Born and Jordan, which is based on the Hilbert space, and (ii) the phase space picture also known as Moyal's Quantization which is based on the quasi- distribution function of Wigner proposed in 1932 and leads to the non-commutative mathematical structure defined by the star product (*). In this work, we present a pedagogical review of points considered basic for the development realised by Feynman. We emphasize Dirac's research line in the search for analogies between the classical and quantum mechanics as well a the postulates formulated by Feynman in the article quoted above. We note that these aspects although important for the comprehension of the path integrals formulation are little known and little divulged of complete form. In order to elucidate the method an example is presented and its relationship to time-dependent Schrödinger equation analyzed.

Neste artigo apresentamos a Teoria das C *-Álgebras. Demonstramos dois importantes resultados que são: o Teorema de Gelfand, que associa toda C *-Álgebra Abeliana às funções contínuas sobre um espaço Hausdorff compacto, e o Teorema de Gelfand-Neumark, que relaciona toda C *-Álgebra não Abeliana aos operadores lineares sobre um espaço de Hilbert. Em seguida, mapeamos a Mecânica Clássica dentro da teoria da C *-Álgebra, obtendo uma prescrição algébrica para os estados e às observáveis clássicas. Ao estendermos essa construção para o caso Quântico, preservando a prescrição algébrica e utilizando o Princípio da Incerteza, obtemos que os estados quânticos devem ser descritos por vetores do Espaço de Hilbert enquanto que os observáveis quânticos são os operadores auto-adjuntos sobre este espaço; discutimos brevemente sobre a Álgebra de Weyl.

In this paper we present the theory of C *-Algebras. We show two major results that are Gelfand's Theorem, which associates every Abelian C *-Algebra as continuous functions on a compact Hausdorff space, and the Gelfand-Neumark Theorem, which relates all non-Abelian C *-Algebra to linear operators on a Hilbert space. Then we map the Classical Mechanics into the C *-Algebra's theory, obtaining an algebraic prescription for classical states and observables. When we extend this construction to the Quantum case, preserving the algebraic prescriptions and using the Uncertainty Principle, we obtain that quantum states must be described by vectors in a Hilbert space while quantum observables are the self-adjoint operators on this space; we briefly discussed the Weyl's Algebra.

Resumen En la primera parte, retomamos testimonios de Fréchet sobre la naturaleza de sus primeros trabajos (1904-1906) en los campos emergentes del Análisis funcional y Análisis general, en relación con su idea de introducir una estructura topológica en un espacio abstracto. En la segunda parte, destacamos la influencia que tuvo en esta idea, el punto de vista algebraico de la época de extender las nociones cantorianas a un espacio abstracto con una estructura de grupo finito. Fréchet supo aprovechar técnicas como el “modo de composición” entre los elementos del espacio, para axiomatizar operaciones y estructuras de la “clase L” con convergencia secuencial, la “clase V” con sistema de vecindades, la “clase E” con “écart” (métrica). Luego se aprovechan nuevos datos históricos para reafirmar la proximidad de las concepciones filosóficas subyacentes a estas investigaciones, con las ideas de Leibniz, específicamente en cuanto al método de “análisis de los principios”. En la tercera parte se estudia la contribución de Hausdorff de 1912 y 1914 al establecimiento de la axiomática de las vecindades para la topología de un espacio abstracto. Teniendo en cuenta las observaciones de Weyl y Bourbaki de que Hausdorff se inspiró para ello en Hilbert, se examina el sistema de axiomas para las vecindades del plano introducido por Hilbert en dos trabajos de 1902 consagrados al problema de la continuidad del espacio. Se exploran las conexiones del “espacio topológico” de Hausdorff basado en las vecindades, con las nociones de métrica, convergencia secuencial y vecindades propuestas años antes por Fréchet. Hausdorff insistió desde el comienzo que la topología del espacio separable tenía las características de generalidad y rigor formal que le permitían adaptarse a las aplicaciones mejor que otras. Se mostrará que todo ello era consistente con los ideales de simplicidad, unidad y economía de pensamiento que Hausdorff había adquirido en sus trabajos filosóficos tempranos.

Abstract In the first part, we remind Fréchet’s testimonies about the approach of his early work (1904-1906) in the emerging fields of Functional Analysis and General Analysis, in relation to his idea of introducing a topological structure in an abstract space. In the second part, we highlight the influence that had on this idea, the algebraic point of view of the time of extending the Cantorian notions to an abstract space with a finite group structure. Fréchet took advantage of techniques such as the “composition mode” between the elements of the space, to axiomatize operations and structures of the “class ” with sequential convergence, the “class ” with neighborhood system, the “class ” with “écart” (metric). Then, new historical data are used to reaffirm the proximity of the philosophical conceptions underlying these investigations with the ideas of Leibniz, specifically with regard to the method of “analysis of principles”. The third part examines the contribution of Hausdorff of 1912 and 1914 to the establishment of the neighborhood axiomatics for the topology of an abstract space. Taking into account the observations of Weyl and Bourbaki that Hausdorff drew on Hilbert, we examine the system of axioms for the neighborhoods of the plane introduced by Hilbert in two 1902 papers devoted to the problem of the space continuity. As regards Fréchet’s influence in Hausdorff, we explore the connections of Hausdorff’s “topological space” based on neighborhoods with the notions of metric, sequential convergence and neighborhoods, proposed years earlier by Fréchet. From the beginning, Hausdorff argued that the topology of the separable space had the characteristics of generality and formal rigor that allowed it to adapt to applications better than others. It is shown that all this was consistent with the ideals of simplicity, unity and economy of thought Hausdorff had acquired in his early philosophical works.

Abstract This paper deals with fractional optimal control for a class of semilinear stochastic equation in Hilbert space setting. To ensure the existence and uniqueness of mild solution, a set of sufficient conditions is constructed. The existence of fractional optimal control for semilinear stochastic system is also discussed. Finally, an example is included to show the applications of the developed theory.

Resumen Este artículo estudia el control óptimo fraccional para una clase de ecuaciones estocásticas semilineales en un contexto de espacios de Hilbert. Para asegurar la existencia y la unicidad de soluciones blandas, construimos un conjunto de condiciones suficientes. La existencia del control óptimo fraccional para sistemas estocásticos semilineales también es discutido. Finalmente, incluimos un ejemplo para mostrar la aplicabilidad de la teoría desarrollada

Resumen Los semigrupos cuánticos de Markov (SCM) son una extensión no conmutativa de los semigrupos de Markov definidos en probabilidad clásica. Ellos representan una evolución sin memoria de un sistema microscopico acorde a las leyes de la física cuántica y a la estructura de los sistemas cuánticos abiertos. Esto significa que la dinámica reducida del sistema principal es descrita por un espacio de Hilbert separable complejo 𝔥 por medio de un semigrupo 𝒯=(𝒯 t ) t ≥0, el cual actúa sobre una subálgebra de von Neumann 𝔐 del álgebra 𝔓(𝔥) de todos los operadores lineales acotados definidos en 𝔥. Por simplicidad, algunas veces asumiremos que 𝔐=(𝔥). El semigrupo 𝓣 corresponde al cuadro de Heisenberg en el sentido que dado cualquier observable x, 𝓣t(x) describe su evolución en el tiempo t. De esta forma, dada una matriz de densidad p, su dinámica (cuadro de Schrödinger) es dada por el semigrupo predual 𝓣 *t (P) , donde tr(P𝓣 t (x))= tr (𝓣 *t (P)x), tr(.) denota la operación traza. En este trabajo ofrecemos una exposición de varios resultados básicos sobre SCM. Además discutimos aplicaciones de SCM en teoría de la información cuántica y computación cuántica.

Abstract Quantum Markov semigroups (SCM) are a non-commutative extension of the Markov semigroups defined in classical probability. They represent an evolution without memory of a microscopic system according to the laws of quantum physics and the structure of open quantum systems. This means that the reduced dynamics of the main system is described by a complex separable Hilbert space 𝔥 by means of a semigroup 𝓣=(𝓣t)t≥0, acting on a von Neumann algebra 𝔓(𝔥) of the linear operators defined on 𝔥. For simplicity, we will sometimes assume that 𝔐=(𝔥). The semigroup 𝓣 corresponds to the Heisenberg picture in the sense that given any observable x, 𝓣t(x) describes its evolution at time t. Thus, given a density matrix p, its dynamics (Schrödinger's picure) is given by the predual semigroup 𝓣*t(≥), where tr(P𝓣 t (x))= tr (𝓣 *t (P)x), tr(.) denote trace of a matrix. In this paper we offer an exposition of several basic results on SCM. We also discuss SCM applications in quantum information theory and quantum computing.

Resumo Os semigrupos quânticos de Markov (SCM) são uma extensão não-comutativa de semigrupos de Markov semigroups definidos na probabilidade clássica. Eles representam uma evolução sem memória de um sistema microscópico acorde com as leis da física quântica e da estrutura de sistemas quânticos abertos. Isto significa que a dinâmica reduzida do sistema principal é descrita por um espaço de Hilbert h complexo separável por um semigroup , que actua sobre um subálgebra de von Neumann M del algebra B(h) de todos os operadores lineares limitados definidos em h. Por simplicidade, por vezes, assumir que M=B(h).. O semigroup T corresponde à imagem Heisenberg no sentido de que, dado qualquer observável x, descreve a evolução no tempo t. Assim, dada uma densidade ρ matriz, dinâmica (caixa de Schrödinger) que é dada pelo semigroupo predual , em que , tr(⋅) denota a operação de traçado. Neste trabalho, oferecemos uma exposição de vários resultados básicos sobre SCM. Além disso, discutimos aplicações de SCM em teoria da informação quântica e computação quântica.

Abstract: In the present paper we characterize, in terms of characters, multiplicative functions, the continuous solutions of some functional equations for mappings defined on a monoid and taking their values in a complex Hilbert space with the Hadamard product. In addition, we investigate a superstability result for these equations.

Let B(H) be the algebra of all bounded linear operators on an infinite-dimensional Hilbert space H. We prove that if Φ is a surjective map on B(H) such that Φ(I) = I + Φ(0) and for every pair T, S ∈ B(H), the operator T - S is singular algebraic if and only if Φ(T) - Φ(S) is singular algebraic, then Φ is either of the form Φ(T) = ATA-1 + Φ(0) or the form Φ(T) = AT*A-1 + Φ(0) where A : H → H is an invertible bounded linear, or conjugate linear, operator.