In mathematics, the Pincherle derivative[1]
of a linear operator
on the vector space of polynomials in the variable x over a field
is the commutator of
with the multiplication by x in the algebra of endomorphisms
End (
is another linear operator
ad
{\displaystyle \operatorname {ad} }
notation, see the article on the adjoint representation) so that This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).
The Pincherle derivative, like any commutator, is a derivation, meaning it satisfies the sum and products rules: given two linear operators
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 This formula generalizes to by induction.
This proves that the Pincherle derivative of a differential operator is also a differential operator, so that the Pincherle derivative is a derivation of
has characteristic zero, the shift operator can be written as by the Taylor formula.
Its Pincherle derivative is then In other words, the shift operators are eigenvectors of the Pincherle derivative, whose spectrum is the whole space of scalars
If T is shift-equivariant, that is, if T commutes with Sh or
is also shift-equivariant and for the same shift
The "discrete-time delta operator" is the operator whose Pincherle derivative is the shift operator