Continuous Linear Map Research Articles

A unified extension of Stieltjes and Poisson transforms to Boehmians

Continuity of the generalized Stieltjes–Poisson transform on compactly supported distributions is proved and convolution theorem is also obtained. Applying the convolution theorem, generalized Stieltjes–Poisson transform is extended as a linear continuous map from compactly supported Boehmians into analytic Boehmians satisfying the convolution theorem.

Integral Representations for a Class of Operators on <em>L</em><sup>1</sup><sub><em>E</em></sub>

Let \((X, \mathcal{A}, \mu)\) be a finite measure space, \(E\) a locally convex Hausdorff space, \(L_{E}^{1}\) the space of functions \(f: X \to E\) which are \(\mu\)-integrable by semi-norms, \(P(\mu, E)\) the space of Pettis integrable functions and \(P_{1}(\mu, E)\) those elements of \(P(\mu, E)\) which are measurable by semi-norms. We prove that a linear continuous mapping \( T: L_{E}^{1} \to E\) is of the form \(T(f)= \int g f d \mu\) (\(g \in L^{\infty}\)) if and only if \( h( T(f))=0\) whenever \( h\circ f=0 \) for any \( f \in L_{E}^{1}, h \in E'\). Similar results are proved for \(P(\mu, E)\) and \(P_{1}(\mu, E)\).

Nearly Jordan ∗‐Homomorphisms between Unital C∗‐Algebras

Let A, B be two unital C∗‐algebras. We prove that every almost unital almost linear mapping h : A → B which satisfies h(3nuy + 3nyu) = h(3nu)h(y) + h(y)h(3nu) for all u ∈ U(A), all y ∈ A, and all n = 0,1, 2, …, is a Jordan homomorphism. Also, for a unital C∗‐algebra A of real rank zero, every almost unital almost linear continuous mapping h : A → B is a Jordan homomorphism when h(3nuy + 3nyu) = h(3nu)h(y) + h(y)h(3nu) holds for all u ∈ I1 (Asa), all y ∈ A, and all n = 0,1, 2, …. Furthermore, we investigate the Hyers‐ Ulam‐Aoki‐Rassias stability of Jordan ∗‐homomorphisms between unital C∗‐algebras by using the fixed points methods.

Some results on $\sigma$-derivations

‎Let $\mathcal{A}$ and $\mathcal{B}$ be two Banach algebras and let‎ ‎$\mathcal{M}$ be a Banach $\mathcal{B}$-bimodule‎. ‎Suppose that‎ ‎$\sigma:\mathcal{A} \rightarrow \mathcal{B}$ is a linear mapping and‎ ‎$d:\mathcal{A} \rightarrow \mathcal{M}$ is a $\sigma$-derivation‎. ‎We‎ ‎prove several results about automatic continuity of‎ ‎$\sigma$-derivations on Banach algebras‎. ‎In addition‎, ‎we define a‎ ‎notion for m-weakly continuous linear mapping and show that‎, ‎under‎ ‎certain conditions‎, ‎$d$ and $\sigma$ are m-weakly continuous‎. ‎Moreover‎, ‎we prove that if $\mathcal{A}$ is commutative and $\sigma‎: ‎\mathcal{A} \rightarrow \mathcal{A}$ is a continuous homomorphism‎ ‎such that $\sigma^{2} = \sigma$ then $\sigma d \sigma (\mathcal{A})‎ ‎\subseteq \sigma(Q(\mathcal{A})) \subseteq rad(\mathcal{A})$‎.

Conditions for the existence of bounded solutions of nonlinear differential and functional differential equations

Let E be a finite-dimensional Banach space, let C0(R; E) be a Banach space of functions continuous and bounded on R and taking values in E; let K:C 0(R ,E) → C 0(R, E) be a c-continuous bounded mapping, let A: E → E be a linear continuous mapping, and let h ∈ C 0(R, E). We establish conditions for the existence of bounded solutions of the nonlinear equation $$ \frac{{dx(t)}}{{dt}} + \left( {Kx} \right)(t)Ax(t) = h(t),\quad t \in \mathbb{R} $$

Approximately zero-product-preserving maps

A continuous linear map T from a Banach algebra A into another B approximately preserves the zero products if ‖T(a)T(b)‖ ≤ α‖a‖‖b‖ (a,b ∈ A, ab = 0) for some small positive α. This paper is mainly concerned with the question of whether any continuous linear surjective map T: A → B that approximately preserves the zero products is close to a continuous homomorphism from A onto B with respect to the operator norm. We show that this is indeed the case for amenable group algebras.

Characterizing Jordan maps on C*-algebras through zero products

AbstractLet A and B be C*-algebras, let X be an essential Banach A-bimodule and let T : A → B and S : A → X be continuous linear maps with T surjective. Suppose that T(a)T(b) + T(b)T(a) = 0 and S(a)b + bS(a) + aS(b) + S(b)a = 0 whenever a, b ε A are such that ab = ba = 0. We prove that then T = wΦ and S = D + Ψ, where w lies in the centre of the multiplier algebra of B, Φ: A → B is a Jordan epimorphism, D: A → X is a derivation and Ψ: A → X is a bimodule homomorphism.

On the Hankel transform for Boehmians

In this paper, we introduce the Hankel transform as a continuous linear map from one space of Boehmian into another and study its operational properties.

Continuity of pseudo-differential operator h μ, a involving Hankel translation and Hankel convolution on some Gevrey spaces

The pseudo-differential operator (p.d.o.) h μ, a associated with the Bessel operator involving the symbol a(x, y) whose derivatives satisfy certain growth conditions depending on some increasing sequences is studied on certain Gevrey spaces. The p.d.o. h μ, a on Hankel translation τ and Hankel convolution of Gevrey functions is a continuous linear mapping into another Gevrey space.

Poisson transform on Boehmians

Convolution theorem for Poisson transform on compactly supported distributions is obtained. Applying the convolution theorem, Poisson transform is extended as a linear continuous map from a suitable Boehmian space into another Boehmian space satisfying the convolution theorem.

Zero product preserving maps on Banach algebras of Lipschitz functions

Let ( K , d ) be a non-empty, compact metric space and α ∈ ] 0 , 1 [ . Let A be either lip α ( K ) or Lip α ( K ) and let B be a commutative unital Banach algebra. We show that every continuous linear map T : A → B with the property that T ( f ) T ( g ) = 0 whenever f , g ∈ A are such that f g = 0 is of the form T = w Φ for some invertible element w in B and some continuous epimorphism Φ : A → B .

A characterization of Fourier transforms

The aim of this paper is to show that, in various situations, the only continuous linear map that transforms a convolution product into a pointwise product is a Fourier transform. We focus on the cyclic groups $\Z/nZ$, the integers $\Z$, the Torus $\T$ and the real line. We also ask a related question for the twisted convolution.

Kontorovich–Lebedev transform for Boehmians

In this paper, we define and study a continuous linear map from one space of Boehmians into another for the Kontorovich–Lebedev transform. The Kontorovich–Lebedev transform is further investigated for compactly supported distributions.

Metrizability of spaces of homomorphisms between metric vector spaces

This note tries to give an answer to the following question: Is there a sufficiently rich class of metric vector spaces such that sufficiently large spaces of continuous linear maps between them are metrizable?

Surjective linear maps preserving local spectra

Let B(X) be the algebra of all bounded linear operators on an infinite dimensional complex Banach space X. We characterize linear surjective and continuous maps on B(X) preserving different local spectral quantities at a nonzero fixed vector.

On the Generalized Stochastic Dirichlet Problem—Part I: The Stochastic Weak Maximum Principle

We treat the stochastic Dirichlet problem \(L\lozenge u = h+\nabla f\) in the framework of white noise analysis combined with Sobolev space methods. The input data and the boundary condition are generalized stochastic processes regarded as linear continuous mappings from the Sobolev space \(W_0^{1,2}\) into the Kondratiev space (S) − 1. The operator L is assumed to be strictly elliptic in divergence form \(L\lozenge u=\nabla(A\lozenge\nabla u+b\lozenge u)+c\lozenge\nabla u+d\lozenge u\). Its coefficients: the elements of the matrix A and of the vectors b, c and d are assumed to be generalized random processes, and the product of two generalized processes, denoted by \(\lozenge\), is interpreted as the Wick product. In this paper we prove the weak maximum principle for the operator L, which will imply the uniqueness of the solution to \(L\lozenge u = h+\nabla f\).

Vectorial exceptional families of elements

Let $${\mathcal{Z}}$$ be an ordered Hausdorff topological vector space with a preorder defined by a pointed closed convex cone $${C \subset {\mathcal Z}}$$ with a nonempty interior. In this paper, we introduce exceptional families of elements w.r.t. C for multivalued mappings defined on a closed convex cone of a normed space X with values in the set $${L(X, {\mathcal Z})}$$ of all continuous linear mappings from X into $${\mathcal{Z}}$$ . In Banach spaces, we prove a vectorial analogue of a theorem due to Bianchi, Hadjisavvas and Schaible. As an application, the C-EFE acceptability of C-pseudomonotone multivalued mappings is investigated.

CONVOLUTION THEOREMS FOR WAVELET TRANSFORM ON TEMPERED DISTRIBUTIONS AND THEIR EXTENSION TO TEMPERED BOEHMIANS

We define a new convolution ⊗ : 𝒮'(ℝ × ℝ+) × 𝒟(ℝ) → 𝒮'(ℝ × ℝ+) and derive the convolution theorems for wavelet transform and dual wavelet transform in the context of tempered distributions. By using the new convolution, we construct a Boehmian space containing the tempered distributions on ℝ × ℝ+. Applying the convolution theorems in the context of tempered distributions, we also extend the wavelet transform and dual wavelet transform between the tempered Boehmian space and the new Boehmian space as linear continuous maps with respect to δ-convergence and Δ-convergence, satisfying the convolution theorems.

Ultraweak Continuity of σ-derivations on von Neumann Algebras

Let σ be a surjective ultraweakly continuous ∗-linear mapping and d be a σ-derivation on a von Neumann algebra \(\mathfrak M\). We show that there are a surjective ultraweakly continuous ∗-homomorphism \(\Sigma:\mathfrak M\to\mathfrak M\) and a Σ-derivation \(D:\mathfrak M\to\mathfrak M\) such that D is ultraweakly continuous if and only if so is d. We use this fact to show that the σ-derivation d is automatically ultraweakly continuous. We also prove the converse in the sense that if σ is a linear mapping and d is an ultraweakly continuous ∗-σ-derivation on \(\mathfrak M\), then there is an ultraweakly continuous linear mapping \(\Sigma:\mathfrak M\to\mathfrak M\) such that d is a ∗-Σ-derivation.

Maps preserving zero products

A linear map $T$ from a Banach algebra $A$ into another $B$ preserves zero products if $T(a)T(b)=0$ whenever $a,b\in A$ are such that $ab=0$. This paper is mainly concerned with the question of whether every continuous linear surjective map $T\colon A\rig