Pincherle derivative

In mathematics, the Pincherle derivative $$T'$$ 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


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

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

Properties
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 $$\operatorname{End}\left( \mathbb{K}[x] \right),$$


 * 1) $$(T + S)^\prime = T^\prime + S^\prime$$;
 * 2) $$(TS)^\prime = T^\prime\!S + TS^\prime$$ 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


 * $$D'= \left({d \over {dx}}\right)' = \operatorname{Id}_{\mathbb K [x]} = 1.$$

This formula generalizes to


 * $$(D^n)'= \left({{d^n} \over {dx^n}}\right)' = nD^{n-1},$$

by induction. This proves that the Pincherle derivative of a differential operator


 * $$\partial = \sum a_n {{d^n} \over {dx^n} } = \sum a_n D^n$$

is also a differential operator, so that the Pincherle derivative is a derivation of $$\operatorname{Diff}(\mathbb K [x])$$.

When $$\mathbb{K}$$ has characteristic zero, the shift operator


 * $$S_h(f)(x) = f(x+h) \,$$

can be written as


 * $$S_h = \sum_{n \ge 0} {{h^n} \over {n!} }D^n$$

by the Taylor formula. Its Pincherle derivative is then


 * $$S_h' = \sum_{n \ge 1} {{h^n} \over {(n-1)!} }D^{n-1} = h \cdot S_h.$$

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 Sh 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 $$\delta' = S_h$$.