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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]

Let H be a real or complex Hilbert space and $ {\mathcal B}(H) $ B (H) the algebra of all bounded linear operators on H. Recall that a map $ \delta :{\mathcal B}(H)\to {\mathcal B}(H) $ δ : B (H) → B (H) is called an inner Jordan $ * $ ∗ -derivation if there exists some $ T\in \mathcal B(H) $ T ∈ B (H) such that $ \delta (A)=AT-TA^* $ δ (A) = AT − T A ∗ for all $ A\in {\mathcal B}(H) $ A ∈ B (H). In this paper, it is proved that inner Jordan $ * $ ∗ -derivations are the only additive maps δ of $ {\mathcal B}(H) $ B (H) with the property that $ \delta (P)=\delta (P)P^*+P\delta (P) $ δ (P) = δ (P) P ∗ + Pδ (P) for all idempotent operators $ P\in {\mathcal B}(H) $ P ∈ B (H) if $ \dim H=\infty $ dim ⁡ H = ∞ , which is satisfied by additive local Jordan $ * $ ∗ -derivations. For the finite dimensional case, additional conditions are required for δ to be an inner Jordan $ * $ ∗ -derivation. As applications, it is shown that, for any given $ C,D\in {\mathcal B}(H) $ C , D ∈ B (H) , δ satisfies $ \delta (A)B^*+B\delta (A)+\delta (B)A^*+A\delta (B)=D $ δ (A) B ∗ + Bδ (A) + δ (B) A ∗ + Aδ (B) = D for all $ A,B\in {\mathcal B}(H) $ A , B ∈ B (H) with AB + BA = C if and only if δ is an inner Jordan $ * $ ∗ -derivation and $ D=\delta (C) $ D = δ (C). Also, several known results are generalized. [ABSTRACT FROM AUTHOR]

In this paper, we explore various transforms associated with a bounded linear operator T on a Hilbert space. These transforms include the Aluthge, λ -Aluthge, Duggal, generalized mean, and λ -mean transforms. Our aim is to investigate the connections between T and these transforms, focusing on aspects such as norm inequalities and numerical ranges, while also highlighting certain essential properties. Furthermore, we aim to determine the conditions under which an operator T coincides in norm with its transformed counterparts through these transformations. Several characterizations and properties are also derived. [ABSTRACT FROM AUTHOR]

Let R be a unital ring containing a nontrivial idempotent. In this article, under a mild condition on R , we prove that if a map δ : R → R satisfies δ (P n (A 1 , A 2 , A 3 , ... , A n)) = ∑ i = 1 n P n (A 1 , ... , A i − 1 , δ (A i) , A i + 1 , ... , A n) for any A 1 , A 2 , A 3 , ... , A n ∈ R with A 1 A 2 A 3 = 0 , then δ (A + B) − δ (A) − δ (B) ∈ Z (R) for any A , B ∈ R . In particular, if R is a von Neumann algebra with no central summands of type I1 or a factor, then δ (x) = d (x) + τ (x) for all x ∈ R , where d : R → R is an additive derivation and τ : R → Z (R) is a map vanishing on each (n − 1) -th commutator p n (A 1 , A 2 , A 3 , ... , A n) with A 1 A 2 A 3 = 0 . [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the boundedness of the fractional type Marcinkiewicz integral operator M α , ρ , m and its higher order commutator M α , ρ , m , b l generated by b ∈ BMO (R n) and M α , ρ , m on the weighted Lebesgue spaces L ω p (R n) . Under assumption that the variable exponents α (·) and q (·) satisfy the log decay at infinity and origin, the authors show that the M α , ρ , m and M α , ρ , m , b l are bounded on the grand variable Herz spaces K ˙ q (·) α (·) , p) , θ (R n) and the grand variable Herz-Morrey spaces M K ˙ p) , θ , q (·) α (·) , λ (R n) , respectively. [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.

The aim of this paper is to establish the boundedness of an θ -type fractional Marcinkiewicz integral M q , ρ , α , θ and its commutator M q , ρ , α , θ , b on weighted Lebesgue spaces L p (ω) , weighted Morrey spaces M p , κ (μ) and generalized weighted Morrey spaces L p , Φ (ω) over RD-spaces satisfying the doubling and reverse doubling conditions. Under assumption that the functions ω and Φ satisfy some certain conditions, the authors prove that the M q , ρ , α , θ is bounded from spaces L p (ω) into spaces L p (ω) , bounded from spaces M p , κ (ω) into spaces M p , κ (ω) , and it is also bounded from spaces L p , Φ (ω) into spaces L p , Φ (ω) , where p ∈ (1 , ∞) , κ ∈ (0 , 1) , ω ∈ A p (μ) and Φ (·) is a non-decreasing and non-negative function defined on (0 , ∞) . Furthermore, by establishing the sharp maximal estimate for the commutator M q , ρ , α , θ , b which is formed by b ∈ BMO (μ) and the M q , ρ , α , θ , the boundedness of the M q , ρ , α , θ , b on spaces L p (ω) , spaces M p , κ (ω) and spaces L p , Φ (ω) is obtained, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the boundedness and compactness of the block dual Toeplitz operators acting on the orthogonal complement of the Fock spaces, and the compactness of the finite sum of two dual Toeplitz operators products. The commutator and semi-commutator induced by the block dual Toeplitz operators are considered. [ABSTRACT FROM AUTHOR]

Let Φ , Ψ be symmetrically norming (s.n.) functions, and L [ μ ] Δ A , B X = def L [ μ ] (Δ A , B ) X = def ∫ R + e - t A X e - t B d μ (t) denotes the Laplace transformer generated by the generalized derivation where μ is a Borel probability measure on If both pairs consist of mutually commuting accretive operators, such that both C - A and D - B are accretive and for some , then L [ μ ] Δ A ∗ , A (I) - L [ μ ] Δ C ∗ , C (I) ⩾ 0 , L [ μ ] Δ B , B ∗ (I) - L [ μ ] Δ D , D ∗ (I) ⩾ 0 and | | C ∗ + C - A ∗ - A (L [ μ ] Δ A , B X - L [ μ ] Δ C , D X) D + D ∗ - B - B ∗ | | Ψ ⩽ | | L [ μ ] Δ A ∗ , A (I) - L [ μ ] Δ C ∗ , C (I) (A X + X B - C X - X D) × L [ μ ] Δ B , B ∗ (I) - L [ μ ] Δ D , D ∗ (I) | | Ψ , holds under any of the following conditions: (a) if (b) if for some p ⩾ 2 , L 2 ( R + , μ) is separable and at least one of pairs (A, C) or (B, D) consists of normal operators, (c) if both pairs (A, C) and (B, D) consist of normal operators. Above, Φ ( p ) denotes (the degree) p-modified s.n. function Φ and Φ ( p ) ∗ is the dual s.n. function for Φ ( p ) . Moreover, the aforementioned inequality is generalized to the iterated perturbations of Laplace transformers, and the alternative inequalities are given for Q norms as well. These inequalities also generalize some previously obtained results. [ABSTRACT FROM AUTHOR]

Let A be a noncommutative prime ring equipped with an involution '∗', and let Q m s (A) be the maximal symmetric ring of quotients of A . Consider the additive maps H and T : A → Q m s (A) . We prove the following under some inevitable torsion restrictions. (a) If m and n are fixed positive integers such that (m + n) T (a 2) = m T (a) a * + n a T (a) for all a ∈ A and (m + n) H (a 2) = m H (a) a * + n a T (a) for all a ∈ A , then H = 0 . (b) If T (a b a) = a T (b) a * for all a , b ∈ A , then T = 0 . Furthermore, we characterize Jordan left τ-centralizers in semiprime rings admitting an anti-automorphism τ. As applications, we find the structure of generalized Jordan ∗-derivations in prime rings and generalize as well as improve all the results of A. Abbasi, C. Abdioglu, S. Ali, M. R. Mozumder (2022). [ABSTRACT FROM AUTHOR]

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]

Let H be a real or complex Hilbert space with the dimension greater than one and B (H) the algebra of all bounded linear operators on H . Assume that δ is a linear mapping from B (H) into itself which is Jordan derivable at a given element Ω ∈ B (H) , in the sense that δ (A ∘ B) = δ (A) ∘ B + A ∘ δ (B) holds for all A , B ∈ B (H) with A ∘ B = Ω , where ∘ denotes the Jordan product A ∘ B = A B + B A . In this paper, we show that if Ω is an arbitrary but fixed nonzero operator, then δ is a derivation; if Ω is a zero operator, then δ is a generalized derivation. [ABSTRACT FROM AUTHOR]

In this paper, we prove the boundedness of the multilinear Littlewood-Paley square operators and their commutators on weighted Morrey spaces, then we give the boundedness and weak-type L log L estimates for the commutators of multilinear Littlewood-Paley g-functions and multilinear Marcinkiewicz integrals on weighted Morrey spaces in the form of corollaries. [ABSTRACT FROM AUTHOR]

We offer a new perspective and some advances on the 1971 Pearcy-Topping problem: is every compact operator a commutator of compact operators? Our goal is to analyze and generalize the 1970’s work in this area of Joel Anderson. We reduce the general problem to a simpler sequence of finite matrix equations with norm constraints, while at the same time developing strategies for counterexamples. Our approach is to ask which compact operators are commutators AB − BA of compact operators A,B; and to analyze the implications of Joel Anderson’s contributions to this problem. By extending the techniques of Anderson, we obtain new classes of operators that are commutators of compact operators beyond those obtained by the second and the fourth author. We also found obstructions to extending Anderson’s techniques to obtain any positive compact operator as a commutator of compact operators. Some of these constraints involve general block-tridiagonal matrix forms for operators and some involve B(H)-ideal constraints. Finally, we provide some necessary conditions for the Pearcy-Topping problem involving singular numbers and B(H)-ideal constraints. [ABSTRACT FROM AUTHOR]

Let A be a noncommutative prime ring equipped with an involution ' ∗ '. Let Q ms (A) be the maximal symmetric ring of quotients of A and C be the extended centroid of A . The objective of this manuscript is to characterize the additive map T : A → Q ms (A) satisfying any one of the following conditions. (i) T (a 2) = a T (a) + η T (a) a ∗ for all a ∈ A , where η ∈ C and η ∉ { 0 , 1 } . (ii) T (a) (a ∗) n + a n T (a) = 0 for all a ∈ A , where n is a fixed positive integer. We also study some results concerning generalized derivations in prime rings with involution. Moreover, all the study in the manuscript is carried out in the light of the theory of functional identities. [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.

We give several basic and spectral properties of classes of closed n-paranormal and closed n ∗ -paranormal operators on dense domains in complex separable Hilbert spaces. We prove that for both of these classes of operators, the null space of (T - μ I) and the range of R (E μ) are identical, where E μ is the Riesz projection with respect to an isolated point μ of the spectrum. We show that they satisfy Weyl's theorem. Certain properties related to the reduced minimum modulus are also established. [ABSTRACT FROM AUTHOR]

Let A be a unital Banach algebra with unit e, M be a Banach A-bimodule, and w ∈ A . In this paper, we characterize those continuous linear maps δ : A → M that satisfy one of the following conditions: δ (a b) = δ (a) b + a δ (b) , 2 δ (w) = δ (a) b + a δ (b) , δ (a b) = δ (a) b + a δ (b) - a δ (e) b , for any a , b ∈ A with a b = b a = w , where w is either a separating point with w ∈ Z (A) or an idempotent. [ABSTRACT FROM AUTHOR]

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 consider bilinear sparse forms intimately related to iterated commutators of a rather general class of operators. We establish Bloom weighted estimates for these forms in the full range of exponents, both in the diagonal and off-diagonal cases. As an application, we obtain new Bloom bounds for commutators of (maximal) rough homogeneous singular integrals and the Bochner–Riesz operator at the critical index. We also raise the question about the sharpness of our estimates. In particular we obtain the surprising fact that even in the case of Calderón–Zygmund operators, the previously known quantitative Bloom bounds are not sharp for the second and higher order commutators. [ABSTRACT FROM AUTHOR]

A Hilbert space operator $ A\in \mathcal {B(H)} $ A ∈ B (H) is a generalized n-projection, $ A\in (G-n-P) $ A ∈ (G − n − P) , if $ {A^*}^n=A $ A ∗ n = A. The product AB of a commuting pair A, B of $ (G-n-P) $ (G − n − P) -operators is a $ (G-n-P) $ (G − n − P) -operator. The converse fails. We prove that if $ \|AB\|=\|A\|\|B\| $ ‖ AB ‖ = ‖ A ‖ ‖ B ‖ and $ \sigma _a(AB)=\sigma _a(A)\sigma _a(B) $ σ a (AB) = σ a (A) σ a (B) , then $ AB\in (G-n-P) $ AB ∈ (G − n − P) implies $ {\tfrac {A}{\|A\|}}, {\tfrac {B}{\|B\|}} $ A ‖ A ‖ , B ‖ B ‖ are $ (G-n-P) $ (G − n − P) if and only if A and B are normal operators. Translated to tensor products (and upon identifying the tensor product $ A\otimes B $ A ⊗ B with the left-right multiplication operator $ {\mathcal {E}}_{A,B^*} $ E A , B ∗ acting on the Hilbert–Schmidt bimodule $ {\mathcal {C}}_2(\mathcal {H}) $ C 2 (H)) this says that $ A\otimes B $ A ⊗ B (resp., $ {\mathcal {E}}_{A,B^*} $ E A , B ∗ ) is a $ (G-n-P) $ (G − n − P) -operator implies $ {\tfrac {A}{\|A\|}}, {\tfrac {B}{\|B\|}} $ A ‖ A ‖ , B ‖ B ‖ are $ (G-n-P) $ (G − n − P) if and only if A and B are normal operators. [ABSTRACT FROM AUTHOR]

Let A be a unital Banach algebra and M be a unital A -bimodule. We show that additive mappings δ , τ : A → M satisfy the identity A − 1 δ (A) + δ (A) A − 1 + A τ (A − 1) + τ (A − 1) A = 0 for all invertible element A in A if and only if there exists a Jordan derivation D such that δ (A) = D (A) + A δ (I) and τ (A) = D (A) − δ (I) A for all A ∈ A . We also characterize additive mappings δ , τ : A → M satisfying the identity δ (A) B + A τ (B) = M for all A , B ∈ A with AB = N, where N is an invertible element in A and M is a fixed element in M. Similar conclusions are obtained for additive mappings from a matrix ring M n (D) into its unital bimodule satisfying any of the above identities, where D is a division ring and char D ≠ 2 . [ABSTRACT FROM AUTHOR]

We show that any nonexpansive derivation on a subalgebra of a dis-jointly complete commutative regular algebra extends up to a derivation on. For an algebra of functions , continuous on a dense open subset of Stone compact X, we establish that the lack of nontrivial derivation is equivalent to σ-distributivity of the Boolean algebra of clopen subsets of X. The field is an arbitrary normed field of charachteristic zero containing a complete non-discrete subfield. Our work is motivated by two seemingly unrelated problems due to Ayupov [2] and Wickstead [32]. [ABSTRACT FROM AUTHOR]

Let N be a maximal discrete nest on an infinite-dimensional separable Hilbert space ℋ , ξ = ∑ n = 1 ∞ e n 2 n be a separating vector for the commutant N ′ , E ξ , be the projection from ℋ onto the subspace [ ℂ ξ ] spanned by the vector ξ, and Q be the projection from K = ℋ ⊕ ℋ ⊕ ℋ onto the closed subspace { (η , η , η) T : η ∈ H } . Suppose that ℒ is the projection lattice generated by the projections ( E ξ 0 0 0 0 0 0 0 0 ) , { ( E 0 0 0 0 0 0 0 0 ) : E ∈ N } , ( I 0 0 0 I 0 0 0 0 ) a n d Q. We show that ℒ is a Kadison-Singer lattice with the trivial commutant. Moreover, we prove that every n-th bounded cohomology group H n (Alg ℒ , B (K)) with coefficients in B (K) is trivial for n ⩾ 1. [ABSTRACT FROM AUTHOR]

In this paper, we solve the system of functional equations { f (x + y) + g (y − x) = 2 f (x) , g (x + y) − f (y − x) = 2 g (y) and we investigate the stability of g-derivations in Banach algebras. [ABSTRACT FROM AUTHOR]

We obtain some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of sharp maximal functions, fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces, where the symbols of the commutators belong to the variable Lipschitz space. A useful tool is that a symbol b belongs a variable Lipschitz space of pointwise type if and only if b belongs to a variable Lipschitz space of integral type under certain assumptions. [ABSTRACT FROM AUTHOR]

If A ( = (A 1 , ... , A d) ), B , S and T are commuting d-tuples of Banach space operators in B (X) d , multiplication A ∙ S , similarly B ∙ T , is defined by coordinatewise multiplication, S commutes with A , B and T , T commutes with B , the pair (A , B) is m-isometric and the pair (S i , T i) is n i -isometric for all 1 ≤ i ≤ d , then the pair (A ∙ S , B ∙ T) is (m + ∑ i = 1 d n i - d) -isometric. [ABSTRACT FROM AUTHOR]

Let U be an associative unital Banach algebra endowed with the Lie product [ x , y ] = x y - y x ( x , y ∈ U ) and create a Lie algebra. In this article, we are going to study the additive maps on U that act at idempotent-products such as centralizers on the Lie structure of U . More precisely, we consider the subsequent condition on an additive map φ on a unital Banach algebra U with a non-trivial idempotent p: x , y ∈ U , x y = p ⟹ φ ([ x , y ]) = [ φ (x) , y ] = [ x , φ (y) ] , and we show under certain conditions that φ (x) = c x + μ (x) for all x ∈ U , where c ∈ Z (U) , μ : U → Z (U) ( Z (U) is the center of U ) is an additive map in which μ ([ x , y ]) = 0 for any x , y ∈ U with x y = p . The obtained results will be used for some Banach algebras, especially, for von Neumann algebras. [ABSTRACT FROM AUTHOR]

Let A be a ∗ -algebra over the complex field C. For any T 1 , T 2 , … , T n ∈ A , define q 1 (T 1) = T 1 , q 2 (T 1 , T 2) = T 1 ⋄ T 2 = T 1 T 2 ∗ + T 2 T 1 ∗ and q n (T 1 , T 2 , … , T n) = q n - 1 (T 1 , T 2 , … , T n - 1) ⋄ T n for all integers n ≥ 2. In this article, it is shown that under certain assumptions a map D : A → A is a nonlinear bi-skew Jordan n-derivation (that is, D satisfies D (q n (T 1 , T 2 , … , T n)) = ∑ i = 1 n q n (T 1 , … , T i - 1 , D (T i) , T i + 1 , … , T n) for all T 1 , T 2 , … , T n ∈ A ) if and only if D is an additive ∗ -derivation. Further, we introduce the notion of a nonlinear generalized bi-skew Jordan n-derivation of a ∗ -algebra and prove that every nonlinear generalized bi-skew Jordan n-derivation G : A → A is of the form G (T) = Z T + D (T) for all T ∈ A , where Z ∈ Z (A) and D : A → A is an additive ∗ -derivation. As applications, we apply our main results to some special classes of ∗ -algebras such as prime ∗ -algebras, factor von Neumann algebras, von Neumann algebras with no central summonds of type I 1 . [ABSTRACT FROM AUTHOR]

Our goal is to characterize the Lipschitz functions by ball Banach function space. We give some necessary and sufficient conditions of the boundedness of the maximal commutators, commutators of the Hardy–Littlewood maximal operators and commutators of the sharp maximal operators when the symbol b belongs to Lipschitz (Campanato) spaces. In light of the results to be obtained, some applications in ball Banach function spaces will also be presented. [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]

Assume that R be a unital prime ring with whose characteristic is not 2 and containing a nontrivial idempotent P. In this paper, it is shown that under certain conditions, ϕ ([ A , B ]) = [ ϕ (A) , B ] where ϕ an additive map on R and AB = P if and only if it has the form ϕ (A) = λ A + h (A) , where h is an additive map into its center vanishing at commutators [ A , B ] with A B = P and λ ∈ Z (R) . An application of our results we characterize generalized Lie derivations on R. Using main result we apply several classical examples of unital prime rings with nontrivial idempotents such as Banach space standard operator algebras and factor von Neumann algebras. [ABSTRACT FROM AUTHOR]

Let b → = (b 1 , b 2 , ... , b m) be a collection of locally integrable functions and T Σ b → the commutator of multilinear singular integral operator T. Denote by L (δ) and L (δ (⋅)) the Lipschitz spaces and the variable Lipschitz spaces, respectively. The main purpose of this paper is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of multilinear commutator T Σ b → in the context of the variable exponent Lebesgue spaces, that is, the authors give the necessary and sufficient conditions for bj (j = 1, 2, ..., m) to be L (δ) or L (δ (⋅)) via the boundedness of multilinear commutator from products of variable exponent Lebesgue spaces to variable exponent Lebesgue spaces. The authors do so by applying the Fourier series technique and some pointwise estimate for the commutators. The key tool in obtaining such pointwise estimate is a certain generalization of the classical sharp maximal operator. [ABSTRACT FROM AUTHOR]

Let (X , d , μ) be a RD-space satisfying both the doubling and reverse doubling conditions. In this setting, the authors first obtain the definitions of a generalized weighted variable exponent Morrey space L ω p (·) , φ (X) and a grand generalized weighted variable exponent Morrey space L ω p (·) , φ , α (X) on RD-spaces. Second, under assumption that φ satisfies some certain condition, the authors prove that the T θ and its commutator generated by b ∈ BMO (X) and the T θ are bounded on spaces L ω p (·) , φ (X) . Finally, via some known results, the boundedness of operators T θ and [ b , T θ ] on spaces L ω p (·) , φ , α (X) is also established. [ABSTRACT FROM AUTHOR]

Coifman, Lions, Meyer and Semmes asked in 1993 whether the Jacobian operator and other compensated compactness quantities map their natural domain of definition onto the real-variable Hardy space H 1 (R n) . We present an axiomatic, Banach space geometric approach to the problem in the case of quadratic operators. We also make progress on the main open case, the Jacobian equation in the plane. [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.

Let X be a Banach space and $ B(X) $ B (X) be the algebra of all bounded linear operators on X. Firstly, we prove that every derivable mapping at a nonzero element in $ B(X) $ B (X) is a derivation. As a corollary, we show that every generalized derivable mapping on $ B(X) $ B (X) is a general derivation. Secondly, we prove that every centralizable mapping on $ B(X) $ B (X) is a centralizer. [ABSTRACT FROM AUTHOR]

A numerical radius inequality due to Kittaneh states that if M is a bounded linear operator on a complex separable Hilbert space, then w 2 (M) ≤ 1 2 M ∗ M + M M ∗ . We prove a numerical radius inequality for finite sums of operators, which is a considerable generalization of Kittaneh inequality: Let M 1 , M 2 , … , M n , N 1 , N 2 , … , N n be operators on a complex separable Hilbert space and let f, g be nonnegative continuous functions on 0 , ∞ which are satisfying the relation f (a) g (a) = a for all a ∈ 0 , ∞. Then for r ≥ 2 , w r ∑ i = 1 n (M i + N i) ≤ 2 r - 2 Z , where Z = ∑ i = 1 n f 2 r M i + f 2 r N i + g 2 r M i ∗ + g 2 r N i ∗ . This inequality is attractive generalization of several recent inequalities. [ABSTRACT FROM AUTHOR]