= Diagonal monomial action of a second-order Euler operator
{title2=$x^2\partial_x^2+y^2\partial_y^2$}
The <monomial> $x^iy^j$ is an <eigenvector> of the differential <linear operator> $x^2\partial_x^2+y^2\partial_y^2$, with <eigenvalue> $i(i-1)+j(j-1)$. Over the real or complex numbers this vanishes exactly for $i,j\in\{0,1\}$. On <polynomials> of bounded <total degree>, the <kernel> is therefore spanned by the permitted members of $1,x,y,xy$, while the <image of a linear map> is spanned by all remaining <monomials>. The <trace> is the sum of those diagonal <eigenvalues>.
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