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A. R. Villena - One of the best experts on this subject based on the ideXlab platform.

  • orthogonally additive polynomials on convolution algebras associated with a compact group
    Journal of Mathematical Analysis and Applications, 2019
    Co-Authors: J. Alaminos, J. Extremera, M L C Godoy, A. R. Villena
    Abstract:

    Abstract Let G be a compact group, let X be a Banach space, and let P : L 1 ( G ) → X be an orthogonally additive, Continuous n-homogeneous polynomial. Then we show that there exists a unique Continuous Linear Map Φ : L 1 ( G ) → X such that P ( f ) = Φ ( f ⁎ ⋯ n ⁎ f ) for each f ∈ L 1 ( G ) . We also seek analogues of this result about L 1 ( G ) for various other convolution algebras, including L p ( G ) , for 1 p ≤ ∞ , and C ( G ) .

  • Zero Jordan product determined Banach algebras
    arXiv: Functional Analysis, 2019
    Co-Authors: J. Alaminos, J. Extremera, Matej Brešar, A. R. Villena
    Abstract:

    A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every Continuous biLinear Map $\varphi\colon A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\varphi(a,b)=\sigma(ab+ba)$ for some Continuous Linear Map $\sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications.

  • orthogonally additive polynomials on convolution algebras associated with a compact group
    arXiv: Functional Analysis, 2018
    Co-Authors: J. Alaminos, J. Extremera, M L C Godoy, A. R. Villena
    Abstract:

    Let $G$ be a compact group, let $X$ be a Banach space, and let $P\colon L^1(G)\to X$ be an orthogonally additive, Continuous $n$-homogeneous polynomial. Then we show that there exists a unique Continuous Linear Map $\Phi\colon L^1(G)\to X$ such that $P(f)=\Phi \bigl(f\ast\stackrel{n}{\cdots}\ast f \bigr)$ for each $f\in L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1< p\le\infty$, and $C(G)$.

  • Orthogonally additive polynomials on Banach function algebras
    Journal of Mathematical Analysis and Applications, 2017
    Co-Authors: A. R. Villena
    Abstract:

    Abstract For a Banach function algebra A, we consider the problem of representing a Continuous d-homogeneous polynomial P : A → X , where X is an arbitrary Banach space, that satisfies the property P ( f + g ) = P ( f ) + P ( g ) whenever f , g ∈ A are such that supp ( f ) ∩ supp ( g ) = ∅ . We show that such a polynomial can be represented as P ( f ) = T ( f d ) ( f ∈ A ) for some Continuous Linear Map T : A → X for a variety of Banach function algebras such as the algebra of Continuous functions C 0 ( Ω ) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lip α ( K ) for any compact metric space K and α ∈ ] 0 , 1 [ , the Figa–Talamanca–Herz algebra A p ( G ) for some locally compact groups G and p ∈ ] 1 , + ∞ [ , the algebras A C ( [ a , b ] ) and B V C ( [ a , b ] ) of absolutely Continuous functions and of Continuous functions of bounded variation on the interval [ a , b ] . In the case where A = C n ( [ a , b ] ) , P can be represented as P ( f ) = ∑ T ( n 1 , … , n d ) ( f ( n 1 ) ⋯ f ( n d ) ) , where the sum is taken over ( n 1 , … , n d ) ∈ Z d with 0 ≤ n 1 ≤ … ≤ n d ≤ n , for appropriate Continuous Linear Maps T ( n 1 , … , n d ) : C n − n d ( [ a , b ] ) → X .

  • Metric versions of Herstein's theorems on Jordan Maps
    Linear & Multilinear Algebra, 2012
    Co-Authors: J. Alaminos, J. Extremera, A. R. Villena
    Abstract:

    Let A be a Banach algebra and let B be an ultraprime Banach algebra. If Φ: A → B is a surjective Continuous Linear Map which tends to satisfy the Jordan multiplicativity condition, then we show that Φ comes near to satisfy either the multiplicativity or the anti-multiplicativity condition. In fact, we give a quantitative estimate of this phenomenon. Furthermore, we estimate how much a Continuous Linear Map Δ: B → B approaches to satisfy the derivation identity in the case when Δ tends to satisfy the Jordan derivation identity

B. B. Waphare - One of the best experts on this subject based on the ideXlab platform.

  • Pseudo-differential type operators and Gevrey spaces
    Asian-European Journal of Mathematics, 2018
    Co-Authors: B. B. Waphare, S. G. Gajbhiv
    Abstract:

    In this paper, the pseudo-differential type operator [Formula: see text] associated with the Bessel type operator [Formula: see text] defined by (2.3) involving the symbol [Formula: see text] whose derivatives satisfy certain growth conditions depending on some increasing sequences, is studied on certain Gevrey spaces. It is shown that the operator [Formula: see text] is a Continuous Linear Map of one Gevrey space into another Gevrey space. A special pseudo-differential type operator called the Gevrey–Hankel type potential is defined and some of its properties are investigated. A variant of [Formula: see text] is also studied.

  • Pseudo-differential operator involving generalized Hankel–Clifford transformation
    Asian-european Journal of Mathematics, 2016
    Co-Authors: P. D. Pansare, B. B. Waphare
    Abstract:

    Pseudo-differential operators (p.d.os) involving generalized Hankel–Clifford transformation associated with the symbol a(x,y) whose derivatives satisfy certain growth condition are defined and the Zemanian type function spaces ℍβ(I) and 𝕊α(I) are introduced. It is shown that p.d.o’s are Continuous Linear Map of the space ℍβ(I) and 𝕊α(I) into itself. Also an Integral representation of p.d.o is obtained.

  • Pseudo-differential operator involving generalized Hankel–Clifford transformation
    Asian-European Journal of Mathematics, 2016
    Co-Authors: P. D. Pansare, B. B. Waphare
    Abstract:

    Pseudo-differential operators (p.d.os) involving generalized Hankel–Clifford transformation associated with the symbol [Formula: see text] whose derivatives satisfy certain growth condition are defined and the Zemanian type function spaces [Formula: see text] and [Formula: see text] are introduced. It is shown that p.d.o’s are Continuous Linear Map of the space [Formula: see text] and [Formula: see text] into itself. Also an Integral representation of p.d.o is obtained.

J. Alaminos - One of the best experts on this subject based on the ideXlab platform.

  • orthogonally additive polynomials on convolution algebras associated with a compact group
    Journal of Mathematical Analysis and Applications, 2019
    Co-Authors: J. Alaminos, J. Extremera, M L C Godoy, A. R. Villena
    Abstract:

    Abstract Let G be a compact group, let X be a Banach space, and let P : L 1 ( G ) → X be an orthogonally additive, Continuous n-homogeneous polynomial. Then we show that there exists a unique Continuous Linear Map Φ : L 1 ( G ) → X such that P ( f ) = Φ ( f ⁎ ⋯ n ⁎ f ) for each f ∈ L 1 ( G ) . We also seek analogues of this result about L 1 ( G ) for various other convolution algebras, including L p ( G ) , for 1 p ≤ ∞ , and C ( G ) .

  • Zero Jordan product determined Banach algebras
    arXiv: Functional Analysis, 2019
    Co-Authors: J. Alaminos, J. Extremera, Matej Brešar, A. R. Villena
    Abstract:

    A Banach algebra $A$ is said to be a zero Jordan product determined Banach algebra if every Continuous biLinear Map $\varphi\colon A\times A\to X$, where $X$ is an arbitrary Banach space, which satisfies $\varphi(a,b)=0$ whenever $a$, $b\in A$ are such that $ab+ba=0$, is of the form $\varphi(a,b)=\sigma(ab+ba)$ for some Continuous Linear Map $\sigma$. We show that all $C^*$-algebras and all group algebras $L^1(G)$ of amenable locally compact groups have this property, and also discuss some applications.

  • orthogonally additive polynomials on convolution algebras associated with a compact group
    arXiv: Functional Analysis, 2018
    Co-Authors: J. Alaminos, J. Extremera, M L C Godoy, A. R. Villena
    Abstract:

    Let $G$ be a compact group, let $X$ be a Banach space, and let $P\colon L^1(G)\to X$ be an orthogonally additive, Continuous $n$-homogeneous polynomial. Then we show that there exists a unique Continuous Linear Map $\Phi\colon L^1(G)\to X$ such that $P(f)=\Phi \bigl(f\ast\stackrel{n}{\cdots}\ast f \bigr)$ for each $f\in L^1(G)$. We also seek analogues of this result about $L^1(G)$ for various other convolution algebras, including $L^p(G)$, for $1< p\le\infty$, and $C(G)$.

  • Metric versions of Herstein's theorems on Jordan Maps
    Linear & Multilinear Algebra, 2012
    Co-Authors: J. Alaminos, J. Extremera, A. R. Villena
    Abstract:

    Let A be a Banach algebra and let B be an ultraprime Banach algebra. If Φ: A → B is a surjective Continuous Linear Map which tends to satisfy the Jordan multiplicativity condition, then we show that Φ comes near to satisfy either the multiplicativity or the anti-multiplicativity condition. In fact, we give a quantitative estimate of this phenomenon. Furthermore, we estimate how much a Continuous Linear Map Δ: B → B approaches to satisfy the derivation identity in the case when Δ tends to satisfy the Jordan derivation identity

  • Zero product preserving Maps on Banach algebras of Lipschitz functions
    Journal of Mathematical Analysis and Applications, 2010
    Co-Authors: J. Alaminos, J. Extremera, A. R. Villena
    Abstract:

    Abstract 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 .

Akhilesh Prasad - One of the best experts on this subject based on the ideXlab platform.

P. D. Pansare - One of the best experts on this subject based on the ideXlab platform.