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Let 풜 {\mathcal{A}} be a unital ∗ {\ast} -algebra over the complex field ℂ {\mathbb{C}} . For any μ 1 , μ 2 , μ 3 , … , μ n ∈ 풜 {\mu_{1},\mu_{2},\mu_{3},\ldots,\mu_{n}\in\mathcal{A}} , a product μ 1 ∘ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 \mu_{1}\circ\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1} is called Jordan product and μ 1 ∙ μ 2 = μ 1 ⁢ μ 2 + μ 2 ⁢ μ 1 ∗ {\mu_{1}\bullet\mu_{2}=\mu_{1}\mu_{2}+\mu_{2}\mu_{1}^{\ast}} is called skew Jordan product. Define P 3 ⁢ ( μ 1 , μ 2 , μ 3 ) = μ 1 ∘ μ 2 ∙ μ 3 P_{3}(\mu_{1},\mu_{2},\mu_{3})=\mu_{1}\circ\mu_{2}\bullet\mu_{3} (called mixed Jordan triple product) and P n ⁢ ( μ 1 , μ 2 , … , μ n ) = μ 1 ∘ μ 2 ∘ ⋯ ∙ μ n P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n})=\mu_{1}\circ\mu_{2}\circ\cdots\bullet\mu% _{n} (called mixed Jordan n-product) for all integer n ≥ 3 {n\geq 3} . In this article, it is shown that a map (called nonlinear mixed Jordan n-derivation) φ : 풜 → 풜 {\varphi:\mathcal{A}\rightarrow\mathcal{A}} satisfies φ ⁢ ( P n ⁢ ( μ 1 , μ 2 , … , μ n ) ) = ∑ i = 1 n P n ⁢ ( μ 1 , … , ν i - 1 , φ ⁢ ( ν i ) , ν i + 1 , … , ν n ) {\varphi(P_{n}(\mu_{1},\mu_{2},\ldots,\mu_{n}))=\sum_{i=1}^{n}P_{n}(\mu_{1},% \ldots,\nu_{i-1},\varphi(\nu_{i}),\nu_{i+1},\ldots,\nu_{n})} for all ν 1 , ν 2 , … , ν n ∈ 풜 {\nu_{1},\nu_{2},\ldots,\nu_{n}\in\mathcal{A}} if and only if φ is an additive ∗ {\ast} -derivation. As applications, our main result is applied to several special classes of unital ∗ {\ast} -algebras such as prime ∗ {\ast} -algebras, factor von Neumann algebras and von Neumann algebras with no central summands of type I 1 {I_{1}} . [ABSTRACT FROM AUTHOR]

Let m ≥ 1 {m\geq 1} , n ≥ 3 {n\geq 3} be fixed integers, ℛ {\mathscr{R}} be a unital ring with a nontrivial idempotent, and let ζ m : ℛ → ℛ {\zeta_{m}:\mathscr{R}\to\mathscr{R}} be a multiplicative Jordan n-higher derivation ( n ≥ 3 ) {(n\geq 3)} . In this paper, we prove that if c ⁢ h ⁢ a ⁢ r ⁢ ( ℛ ) ≠ 2 ⁢ 푎푛푑 ≠ n - 1 {char(\mathscr{R})\neq 2~{}\textit{and}~{}\neq n-1} , then ζ m {\zeta_{m}} is an additive Jordan n-higher derivation, more precisely, ζ m ⁢ ( a ) = D m ⁢ ( a ) + π m ⁢ ( a ) {\zeta_{m}(a)=D_{m}(a)+\pi_{m}(a)} for all a ∈ ℛ {a\in\mathscr{R}} , where D m : ℛ → ℛ {D_{m}:\mathscr{R}\to\mathscr{R}} is an additive higher derivation and π m : ℛ → ℛ {\pi_{m}:\mathscr{R}\to\mathscr{R}} is an additive singular Jordan higher derivation. [ABSTRACT FROM AUTHOR]

If T1{{\mathbb{T}}}_{1} and T2{{\mathbb{T}}}_{2} are commuting dd-tuples of Hilbert space operators in B(ℋ)dB{\left({\mathcal{ {\mathcal H} }})}^{d} such that (T1*⊗I+I⊗T2*,T1⊗I+I⊗T2)\left({{\mathbb{T}}}_{1}^{* }\otimes I+I\otimes {{\mathbb{T}}}_{2}^{* },{{\mathbb{T}}}_{1}\otimes I+I\otimes {{\mathbb{T}}}_{2}) is strictly mm-isometric (resp., mm-symmetric) for some positive integer mm, then there exist a scalar dd-tuple λ\lambda and positive integers mi{m}_{i}, 1≤i≤21\le i\le 2, such that m=m1+2m2−2m={m}_{1}+2{m}_{2}-2, (T1*+λ¯,T1+λ)\left({{\mathbb{T}}}_{1}^{* }+\overline{\lambda },{{\mathbb{T}}}_{1}+\lambda ) is m1{m}_{1} isometric, and T2−λ{{\mathbb{T}}}_{2}-\lambda is m2{m}_{2}-nilpotent (resp., m=m1+m2−1m={m}_{1}+{m}_{2}-1, (T1*+λ¯,T1+λ)\left({{\mathbb{T}}}_{1}^{* }+\overline{\lambda },{{\mathbb{T}}}_{1}+\lambda ) is m1{m}_{1}-symmetric and (T2*−λ¯,T2−λ)\left({{\mathbb{T}}}_{2}^{* }-\overline{\lambda },{{\mathbb{T}}}_{2}-\lambda ) is m2{m}_{2} symmetric).

In this work, we present necessary and sufficient conditions for the boundedness of the commutators generated by multilinear fractional maximal operators on the products of Morrey spaces when the symbol belongs to Lipschitz spaces.

Let 풜 be a standard operator algebra containing the identity operator I on an infinite dimensional complex Hilbert space ℋ which is closed under adjoint operation. Suppose that ϕ : 풜 → 풜 is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive ∗ -derivation. [ABSTRACT FROM AUTHOR]

In this study, we solve the system of additive functional equations: h(x+y)=h(x)+h(y),g(x+y)=f(x)+f(y),2fx+y2=g(x)+g(y),\left\{\begin{array}{l}h\left(x+y)=h\left(x)+h(y),\\ g\left(x+y)=f\left(x)+f(y),\\ 2f\left(\frac{x+y}{2}\right)=g\left(x)+g(y),\end{array}\right. and we investigate the stability of (homomorphism, derivation)-systems in Banach algebras.

According to Posner's second theorem, a prime ring is forced to be commutative if a nonzero centralizing derivation exists on it. In this article, we extend this result to prime rings with antiautomorphisms and nonzero skew derivations. Additionally, a case is shown to demonstrate that the restrictions placed on the theorems' hypothesis were not unnecessary. [ABSTRACT FROM AUTHOR]

In this paper, we apply the Cădariu–Radu method derived from the Diaz–Margolis theorem to investigate existence, uniqueness approximation of Ξ-Hilfer fractional differential equations, and Hypergeometric stability for both finite and infinite domains. An example is given to illustrate the main result for a fractional system. [ABSTRACT FROM AUTHOR]

Let (X,d,μ)\left({\mathcal{X}},d,\mu ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space Mpu(μ){M}_{p}^{u}\left(\mu ), where 1≤p

The main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise estimates for the commutators. The key tool in obtaining such a pointwise estimate is a certain generalization of the classical sharp maximal operator.

The main goal of this paper is to characterize Jordan two-sided centralizers, Jordan centralizers and related maps on triangular rings without identity. As an application of our main theorem, we characterize Jordan generalized derivations on triangular rings. Precisely, we prove that every Jordan generalized derivation on a triangular ring is a two-sided generalized derivation. As consequences, and apart from proving the other results, many known theorems can be either generalized or deduced. [ABSTRACT FROM AUTHOR]

Let 프 be a standard operator algebra on a complex Banach space 픛 , dim ⁡ 픛 > 1 , and p n ⁢ (T 1 , T 2 , ... , T n) the (n - 1) th-commutator of elements T 1 , T 2 , ... , T n ∈ 프 . Then every map ξ : 프 → 프 (not necessarily linear) satisfying ξ ⁢ (p n ⁢ (T 1 , T 2 , ... , T n)) = ∑ i = 1 n p n ⁢ (T 1 , T 2 , ... , T i - 1 , ξ ⁢ (T i) , T i + 1 , ... , T n) for all T 1 , T 2 , ... , T n ∈ 프 is of the form ξ = Ω + Γ , where Ω : 프 → 프 is an additive derivation and Γ : 프 → ℂ ⁢ I is a map that vanishes at each (n - 1) th-commutator p n ⁢ (T 1 , T 2 , ... , T n) for all T 1 , T 2 , ... , T n ∈ 프 . In addition, if the map ξ is linear and satisfies the above relation, then there exist an operator S ∈ 프 and a linear map Γ : 프 → ℂ ⁢ I satisfying Γ ⁢ (p n ⁢ (T 1 , T 2 , ... , T n)) = 0 for all T 1 , T 2 , ... , T n ∈ 프 , such that ξ ⁢ (T) = [ T , S ] + Γ ⁢ (T) for all T ∈ 프 . [ABSTRACT FROM AUTHOR]

In this article, we solve the system of additive functional equations: 2f(x+y)−g(x)=g(y),g(x+y)−2f(y−x)=4f(x)\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}2f\left(x+y)-g\left(x)=g(y),\\ g\left(x+y)-2f(y-x)=4f\left(x)\end{array}\right. and prove the Hyers-Ulam stability of the system of additive functional equations in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of ff-hom-ders in Banach algebras.

In this article, the authors consider the commutators of strongly singular Calderón-Zygmund operator with Lipschitz functions. A sufficient condition is given for the boundedness of the commutators from Lebesgue spaces Lp(Rn){L}^{p}\left({{\mathbb{R}}}^{n}) to certain Campanato spaces Cp,β(Rn){C}^{p,\beta }\left({{\mathbb{R}}}^{n}).

Let A{\mathcal{A}} be a ∗\ast -algebra, ℳ{\mathcal{ {\mathcal M} }} be a ∗\ast -A{\mathcal{A}}-bimodule, and δ\delta be a linear mapping from A{\mathcal{A}} into ℳ{\mathcal{ {\mathcal M} }}. δ\delta is called a ∗\ast -derivation if δ(AB)=Aδ(B)+δ(A)B\delta \left(AB)=A\delta \left(B)+\delta \left(A)B and δ(A∗)=δ(A)∗\delta \left({A}^{\ast })=\delta {\left(A)}^{\ast } for each A,BA,B in A{\mathcal{A}}. Let GG be an element in A{\mathcal{A}}, δ\delta is called a ∗\ast -antiderivable mapping at GG if AB∗=G⇒δ(G)=B∗δ(A)+δ(B)∗AA{B}^{\ast }=G\Rightarrow \delta \left(G)={B}^{\ast }\delta \left(A)+\delta {\left(B)}^{\ast }A for each A,BA,B in A{\mathcal{A}}. We prove that if A{\mathcal{A}} is a C∗{C}^{\ast }-algebra, ℳ{\mathcal{ {\mathcal M} }} is a Banach ∗\ast -A{\mathcal{A}}-bimodule and GG in A{\mathcal{A}} is a separating point of ℳ{\mathcal{ {\mathcal M} }} with AG=GAAG=GA for every AA in A{\mathcal{A}}, then every ∗\ast -antiderivable mapping at GG from A{\mathcal{A}} into ℳ{\mathcal{ {\mathcal M} }} is a ∗\ast -derivation. We also prove that if A{\mathcal{A}} is a zero product determined Banach ∗\ast -algebra with a bounded approximate identity, ℳ{\mathcal{ {\mathcal M} }} is an essential Banach ∗\ast -A{\mathcal{A}}-bimodule and δ\delta is a continuous ∗\ast -antiderivable mapping at the point zero from A{\mathcal{A}} into ℳ{\mathcal{ {\mathcal M} }}, then there exists a ∗\ast -Jordan derivation Δ\Delta from A{\mathcal{A}} into ℳ♯♯{{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } and an element ξ\xi in ℳ♯♯{{\mathcal{ {\mathcal M} }}}^{\sharp \sharp } such that δ(A)=Δ(A)+Aξ\delta \left(A)=\Delta \left(A)+A\xi for every AA in A{\mathcal{A}}. Finally, we show that if A{\mathcal{A}} is a von Neumann algebra and δ\delta is a ∗\ast -antiderivable mapping (not necessary continuous) at the point zero from A{\mathcal{A}} into itself, then there exists a ∗\ast -derivation Δ\Delta from A{\mathcal{A}} into itself such that δ(A)=Δ(A)+Aδ(I)\delta \left(A)=\Delta \left(A)+A\delta \left(I) for every AA in A{\mathcal{A}}.

We investigate the additivity and multiplicativity of centrally extended higher derivations and show that every centrally extended higher derivation of a semiprime ring with no nonzero central ideals is a higher derivation. Moreover, we study preservation of the center of the ring by a centrally extended higher derivation.

Let ℋ{\mathcal{ {\mathcal H} }} be a complex Hilbert space and ℬ(ℋ){\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) denotes the algebra of all bounded linear operators acting on ℋ{\mathcal{ {\mathcal H} }}. In this paper, we present some new pairs of generalized finite operators. More precisely, new pairs of operators (A,B)∈ℬ(ℋ)×ℬ(ℋ)\left(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}) satisfying: ∥AX−XB−I∥≥1,for allX∈ℬ(ℋ).\parallel AX-XB-I\parallel \ge 1,\hspace{1.0em}\hspace{0.1em}\text{for all}\hspace{0.1em}\hspace{0.33em}X\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}). An example under which the class of such operators is not invariant under similarity orbit is given. Range kernel orthogonality of generalized derivation is also studied.

The purpose of this paper is to study the commutativity of a prime ring ℛ with anti-automorphism ψ and generalized derivation (ℱ , ξ) satisfying certain algebraic identities. Furthermore, we describe the structure of any additive map which is ψ-centralizing on a 2-torsion free prime ring with anti-automorphism ψ. We also provide examples to show that various conditions imposed on the hypotheses of our results are essential. [ABSTRACT FROM AUTHOR]

A new class of operators, larger than ∗\ast -finite operators, named generalized ∗\ast -finite operators and noted by Gℱ∗(ℋ){{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }}) is introduced, where: Gℱ∗(ℋ)={(A,B)∈ℬ(ℋ)×ℬ(ℋ):∥TA−BT∗−λI∥≥∣λ∣,∀λ∈C,∀T∈ℬ(ℋ)}.{{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}. Basic properties are given. Some examples are also presented.

The characterization of the points in Cp:1≤p

We study the range-kernel weak orthogonality of certain elementary operators induced by non-normal operators, with respect to usual operator norm and the Von Newmann-Schatten pp-norm (1≤p

The question whether every operator on infinite-dimensional Hilbert space 퐻 has a nontrivial invariant subspace or a nontrivial hyperinvariant subspace is one of the most difficult problems in operator theory. This problem is open for more than half a century. A subnormal operator has a nontrivial invariant subspace, but the existence of nontrivial invariant subspace for a hyponormal operator 푇 still open. In this paper we give an affirmative answer of the existence of a nontrivial hyperinvariant subspace for a hyponormal operator. More generally, we show that a large classes of operators containing the class of hyponormal operators have nontrivial hyperinvariant subspaces. Finally, every generalized scalar operator on a Banach space 푋 has a nontrivial invariant subspace. [ABSTRACT FROM AUTHOR]

In this paper, we study the Hyers-Ulam-Rassias stability of (m,n)(m,n)-Jordan derivations. As applications, we characterize (m,n)(m,n)-Jordan derivations on C⁎{C}^{\ast }-algebras and some non-self-adjoint operator algebras.

The structure of Jordan centralizer maps is investigated on trivial extension algebras. One may obtain some conditions under which a Jordan centralizer map on a trivial extension algebra is a centralizer map. As an application, we characterize the Jordan centralizer map on a triangular algebra.

In 1977, Colville, Davis, and Keimel [Positive derivations on f-rings, J. Aust. Math. Soc. Ser. A 23 1977, 3, 371–375] proved that a positive derivation on an Archimedean f-algebra A has its range in the set of nilpotent elements of A. The main objective of this paper is to obtain a generalization of the above Colville, Davis and Keimel result to general derivations. Moreover, we give a new version of the Singer–Wermer conjecture for the class of second-order derivations acting on uniformly complete almost f-algebras. [ABSTRACT FROM AUTHOR]

Let 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras. [ABSTRACT FROM AUTHOR]