Type of derivative of a linear operator
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 . That is, 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 and belonging to
- ;
- where is the
composition of operators.
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 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 , then we also have , so that is also shift-equivariant and for the same shift .
The "discrete-time delta operator"
is the operator
whose Pincherle derivative is the shift operator .
See also
References
External links