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Algebra i Analiz, 2012, Volume 24, Issue 4, Pages 137–155 (Mi aa1295)  

This article is cited in 1 scientific paper (total in 1 paper)

Research Papers

An operator equation characterizing the Laplacian

H. Königa, V. Milmanb

a Mathematisches Seminar, Universität Kiel, Kiel, Germany
b School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel

Abstract: The Laplace operator on $\mathbb R^n$ satisfies the equation
$$ \Delta(fg)(x)=(\Delta f)(x)g(x)+f(x)(\Delta g)(x)+2\langle f'(x),g'(x)\rangle $$
for all $f,g\in C^2(\mathbb R^n,\mathbb R)$ and $x\in\mathbb R^n$. In the paper, an operator equation generalizing this product formula is considered. Suppose $T\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)$ and $A\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R^n)$ are operators satisfying the equation
\begin{equation} T(fg)(x)=(Tf)(x)g(x)+f(x)(Tg)(x)+\langle(Af)(x),(Ag)(x)\rangle \tag{1} \end{equation}
for all $f,g\in C^2(\mathbb R^n,\mathbb R)$ and $x\in\mathbb R^n$. Assume, in addition, that $T$ is $O(n)$-invariant and annihilates the affine functions, and that $A$ is nondegenerate. Then $T$ is a multiple of the Laplacian on $\mathbb R^n$, and $A$ a multiple of the derivative,
$$ (Tf)(x)=\frac{d(\|x\|)^2}2(\Delta f)(x),\quad (Af)(x)=d(\|x\|)f'(x), $$
where $d\in C(\mathbb R_+,\mathbb R)$ is a continuous function. The solutions are also described if $T$ is not $O(n)$-invariant or does not annihilate the affine functions. For this, all operators $(T,A)$ satisfying (1) for scalar operators $A\colon C^2(\mathbb R^n,\mathbb R)\to C(\mathbb R^n,\mathbb R)$ are determined. The map $A$, both in the vector and the scalar case, is closely related to $T$ and there are precisely three different types of solution operators $(T,A)$.
No continuity or linearity requirement is imposed on $T$ or $A$.

Keywords: Laplace operator, second order Leibniz rule, operator functional equations.

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English version:
St. Petersburg Mathematical Journal, 2013, 24:4, 631–644

Bibliographic databases:

Received: 01.11.2011
Language:

Citation: H. König, V. Milman, “An operator equation characterizing the Laplacian”, Algebra i Analiz, 24:4 (2012), 137–155; St. Petersburg Math. J., 24:4 (2013), 631–644

Citation in format AMSBIB
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\by H.~K\"onig, V.~Milman
\paper An operator equation characterizing the Laplacian
\jour Algebra i Analiz
\yr 2012
\vol 24
\issue 4
\pages 137--155
\mathnet{http://mi.mathnet.ru/aa1295}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3088010}
\zmath{https://zbmath.org/?q=an:1273.47035}
\elib{http://elibrary.ru/item.asp?id=20730169}
\transl
\jour St. Petersburg Math. J.
\yr 2013
\vol 24
\issue 4
\pages 631--644
\crossref{https://doi.org/10.1090/S1061-0022-2013-01257-X}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84878627261}


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    This publication is cited in the following articles:
    1. Proc. Steklov Inst. Math., 280 (2013), 191–207  mathnet  crossref  crossref  mathscinet  isi  elib
  • Алгебра и анализ St. Petersburg Mathematical Journal
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