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 Let B(X) be the space of all bounded linear operators on complex Banach space X. For A ∈ B(X), we denote by F(A) the subspace of all fixed points of A. In this paper, we study and characterize all surjective maps φ on B(X) satisfying F(φ(T)φ(A) + φ(A)φ(T)) = F(T A + AT) for all A, T ∈ B(X).

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.

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 Let X and Y be two complex Banach spaces, and let B(X) denotes the algebra of all bounded linear operators on X. We characterize additive maps from B(X) onto B(Y ) compressing the pseudospectrum subsets Δ ϵ (.), where Δ ϵ (.) stands for any one of the spectral functions σ ϵ (.), σ l ϵ (.) and σ r ϵ (.) for some ϵ > 0. We also characterize the additive (resp. non-linear) maps from B(X) onto B(Y) preserving the pseudospectrum σ ϵ (.) of generalized products of operators for some ϵ > 0 (resp. for every ϵ > 0).

Abstract We carry out characterizations with techniques provided by the local spectral theory of bounded linear operators T ∈ L(X), X infinite dimensional complex Banach space, which verify property (V Π ) introduced by Sanabria et al. (Open Math. 16(1) (2018), 289-297). We also carry out the study for polaroid operators and Drazin invertible operators that verify the property mentioned above.

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.

Abstract We study common local spectral properties for bounded linear operators A ∈ ℒ(X,Y) and B,C ∈ ℒ (Y,X) such that A(BA) 2 =ABACA=ACABA=(AC) 2 A. We prove that AC and BA share the single valued extension property, the Bishop property (β), the property (β ε ), the decomposition property (δ) and decomposability. Closedness of analytic core and quasinilpotent part are also investigated. Some applications to Fredholm operators are given.

Abstract Let B(X) be the algebra of all bounded linear operators in a complex Banach space X. For A ∈ B(X) let F (A) be the subspace of fixed point of A. For an integer k ≥ 2, let (i 1 , .., i m ) be a finite sequence with terms chosen from {1, · · · , k}, and assume at least one of the terms in (i 1 , · · · , i m ) appears exactly once. The generalized product of k operators A 1 , ..., A k ∈ B(X) is defined by A 1 ∗ A 2 ∗ · · · ∗ A k = A i ₁ A i ₂ · · · A i m , and includes the usual product and the triple product. We characterize the form of maps from B(X) onto itself satisfying F (ϕ(A 1 ) ∗ · · · ∗ ϕ(A k )) = F (A 1 ∗ · · · ∗ A k ) for all A 1 , · · · , A k ∈ B(X).

Abstract In this paper, we give some characterizations of the left and right generalized Drazin invertible bounded operators in Banach spaces by means of the single-valued extension property (SVEP). In particular, we show that a bounded operator is left (resp. right) generalized Drazin invertible if and only if admits a generalized Kato decomposition and has the SVEP at 0 (resp. it admits a generalized Kato decomposition and its adjoint has the SVEP at 0. In addition, we prove that both of the left and the right generalized Drazin operators are invariant under additive commuting finite rank perturbations. Furthermore, we investigate the transmission of some local spectral properties from a bounded linear operator, as the SVEP, Dunford property (C), and property (β), to its generalized Drazin inverse.

Abstract Let X be a Banach space and let B(X) be the Banach algebra of all bounded linear operators on X. We characterise surjective (not necessarily linear or additive) maps ϕ : B(X) → B(X) such that F(ϕ (A)◇ ϕ (B)) = F(A ◇ B) for all A,B ∈ B(X) where F(A) denotes any of R(A) or N(A), anda ◇ B denotes any binary operations A−B, AB and ABA for all A,B ∈B(X).

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.

Let Ti be a complex Hilbert space and B(TT) the algebra of all bounded linear operators on H. We give the concrete forms of surjec-tive continue unital linear maps from B(TT) onto itself that preserves G-partial-isometric operators.

The main purpose of this paper is to prove the following result, which is related to a classical result of Chernoff. Let X be a real or complex Banach space, let L (X) be the algebra of all bounded linear operators on X and let A(X) C L (X) be a standard operator algebra. Suppose there exists a linear mapping D : A (X) ͢ L (X) satisfying the relation 2D (An) = D (An-1) A+An-1 D (A) + D (A) An-1+ AD (An-1) for all A e A (X), where n > 2 is some fixed integer. In this case D is of the form D (A) = [A, B] for all A e A (X) and some fixed B e L (X), which means that D is a linear derivation. In particular, D is continuous.

El propósito principal de este artículo es probar el siguiente resultado, el cual se relaciona a un resultado clásico de Chernoff. Sea X un espacio de Banach real o complejo, sea L (X) el álgebra de todos los operadores lineales acotados en X y sea A (X) C L (X) una álgebra de operadores estándar. Supongamos que existe una aplicación lineal D : A (X) ͢ L (X) satisfaciendo la relación 2D (An) = D (An-1) A + An-1D (A) + D (A) An-1 + AD (An-1) para todo A e A (X), donde n > 2 es algún entero fijo. En este caso D es de la forma D (A) = [A, B] para todo A e A (X) y algún B e L (X) fijo, lo que significa que D es una derivación lineal. En particular, D es continua.

This paper deals with discrete almost periodic linear operators in the space of bounded sequences. We study the invertibility of such operators in that space, as well as in the space of almost periodic sequences. One of main results is a discrete version of wellknown First Favard Theorem, and is based on the notion of the envelope of an almost periodic operator. Another result is restricted to finite order operators. It characterizes the invertibility in therms of the operator in question only.

Este trabajo trata de operadores lineales discretos casi periódicos en el espacio de las secuencias acotadas. Estudiamos la invertibilidad de dichos operadores en ese espacio, así como en el espacio de secuencias casi periódicas. Uno de los resultados principales es una versión discreta del conocido Primer Teorema de Favard, y se basa en la noción de la envolvente de un operador casi periódico. Otro resultado se restringe a los operadores de orden finito. Se caracteriza la invertibilidad solamente en términos del operador en cuestión.