In mathematics, the Pincherle derivative T' of a linear operator 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. That is, T' is another linear operator (for the origin of the notation, see the article on the adjoint representation) so that 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 One also has where 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. When \mathbb{K} 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 \mathbb{K}. If T is shift-equivariant, that is, if T commutes with Sh or [T,S_h] = 0, then we also have, so that T' is also shift-equivariant and for the same shift h. The "discrete-time delta operator" is the operator whose Pincherle derivative is the shift operator.