相关论文: Derivative of Map of Banach algebra

Let $A$, $B$ be Banach $D$-algebras. The map $f:A\rightarrow B$ is called differentiable on the set $U\subset A$, if at every point $x\in U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =\frac{d f(x)}{d x}\circ dx +o(dx)$$…

In the paper I considered definition and structure of linear mapping of Banach algebra over commutative ring. Based on this definition I explore derivative of continuous mapping.

I consider differential of mapping $f$ of continuous division ring as linear mapping the most close to mapping $f$. Different expressions which correspond to known deffinition of derivative are supplementary. I explore the Gateaux…

We prove that for every function $f:X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is Gateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$…

Let $X$ be a separable Banach space, $Y$ a Banach space and $f: X \to Y$ an arbitrary mapping. Then the following implication holds at each point $x \in X$ except a $\sigma$-directionally porous set: If the one-sided Hadamard directional…

We define the tangential derivative, a notion of directional derivative which is invariant under diffeomorphisms. In particular this derivative is invariant under changes of chart and is thus well-defined for functions defined on a…

Let $\mathcal A$ be a von Neumann algebra and $\mathcal M$ be a Banach $\mathcal A-$module. It is shown that for every homomorphisms $\sigma, \tau$ on $\mathcal A$, every bounded linear map $f:\mathcal A\to \mathcal M$ with property that…

We investigate the differentiability properties of real-valued quasiconvex functions f defined on a separable Banach space X. Continuity is only assumed to hold at the points of a dense subset. If so, this subset is automatically residual.…

Let $f$ be a Lipschitz map from a subset $A$ of a stratified group to a Banach homogeneous group. We show that directional derivatives of $f$ act as homogeneous homomorphisms at density points of $A$ outside a $\sigma$-porous set. At…

Given a function $f:(0,\infty)\rightarrow\RR$ and a positive semidefinite $n\times n$ matrix $P$, one may define a trace functional on positive definite $n\times n$ matrices as $A\mapsto \Tr(Pf(A))$. For differentiable functions $f$, the…

We show that the derivatives in the sense of Fr\'echet and G\^ateaux can be viewed as derivatives oriented towards a star convex set with the origin as center. The resulting oriented differential calculus extends the mean value theorem, the…

We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…

Differentiation in mathematical analysis is commonly built by using {\epsilon}-{\delta}-language. This approach also works similarly for defining continuity, Gateaux (directional) derivative and Frechet derivative in normed vector spaces,…

This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear…

Let $\mathcal{M}$ be a Banach bimodule over an associative Banach algebra $\mathcal{A}$, and let $F: \mathcal{A}\to \mathcal{M}$ be a linear mapping. Three main uses of the term \emph{generalized derivation} are identified in the available…

We prove that a locally bounded and differentiable in the sense of Gateaux function given in a finite-dimensional commutative Banach algebra over the complex field is also differentiable in the sense of Lorch.

Let $n\in \Bbb N-\{1\},$ and let $A$ be a Banach algebra. An additive map $D: A\to A$ is called n-Jordan derivation if $$D(a^n)=D(a)a^{n-1}+aD(a)a^{n-2}+...+a^{n-2}D(a)a+a^{n-1}D(a),$$ for all $a \in {A}$. Using fixed point methods, we…

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each {\epsilon}>0,…

A type of fractional derivative, referred to as \alpha-derivative, is studied. The \alpha-derivative of fractional type obeys Leibnitz rule. Based on the definition of \alpha-derivative the operations of analysis and differential geometry…

Let $\cal A$ and $\cal B$ be Banach algebras. A linear map $T:{\cal A} \rightarrow {\cal B}$ is called separating or disjointness preserving if $ab=0$ implies $Ta\;Tb = 0$ for all $a,b\in {\cal A}$. In this paper, we study a new class of…