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

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

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

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

Properties

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The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators$S$ and $T$ belonging to $\operatorname {End} \left(\mathbb {K} [x]\right),$

$(T+S)^{\prime }=T^{\prime }+S^{\prime }$;

$(TS)^{\prime }=T^{\prime }\!S+TS^{\prime }$ where $TS=T\circ S$ is the composition of operators.

One also has $[T,S]^{\prime }=[T^{\prime },S]+[T,S^{\prime }]$ where $[T,S]=TS-ST$ is the usual Lie bracket, which follows from the Jacobi identity.

The usual derivative, D = d/dx, is an operator on polynomials. By straightforward computation, its Pincherle derivative is

In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars $\mathbb {K}$.

If T is shift-equivariant, that is, if T commutes with S_{h} or $[T,S_{h}]=0$, then we also have $[T',S_{h}]=0$, so that $T'$ is also shift-equivariant and for the same shift $h$.

The "discrete-time delta operator"

$(\delta f)(x)={{f(x+h)-f(x)} \over h}$

is the operator

$\delta ={1 \over h}(S_{h}-1),$

whose Pincherle derivative is the shift operator $\delta '=S_{h}$.