Pincherle derivative

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In mathematics, the Pincherle derivative^[1] ${\displaystyle T'}$ of a linear operator ${\displaystyle T:\mathbb {K} [x]\to \mathbb {K} [x]}$ on the vector space of polynomials in the variable x over a field ${\displaystyle \mathbb {K} }$ is the commutator of ${\displaystyle T}$ with the multiplication by x in the algebra of endomorphisms ${\displaystyle \operatorname {End} (\mathbb {K} [x])}$. That is, ${\displaystyle T'}$ is another linear operator ${\displaystyle T':\mathbb {K} [x]\to \mathbb {K} [x]}$

${\displaystyle T':=[T,x]=Tx-xT=-\operatorname {ad} (x)T,\,}$

(for the origin of the ${\displaystyle \operatorname {ad} }$ notation, see the article on the adjoint representation) so that

${\displaystyle T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad \forall p(x)\in \mathbb {K} [x].}$

This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).