# Theta operator

In mathematics, the theta operator is a differential operator defined by[1][2]

$\theta = z {d \over dz}$

This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in z:

$\theta (z^k) = k z^k,\quad k=0,1,2,\dots$

In n variables the homogeneity operator is given by

$\theta = \sum_{k=1}^n x_k \frac{\partial}{\partial x_k}.$

As in one variable, the eigenspaces of θ are the spaces of homogeneous polynomials.