  Given a *-homomorphism $\sigma: C(M)\to \mathscr{L}(\mathcal{H})$ on a
Hilbert space $\mathcal{H}$ for a compact metric space $M$, a projection $P$
onto a subspace $\mathcal{P}$ in $\mathcal{H}$ is said to be essentially normal
relative to $\sigma$ if $[\sigma(\varphi),P]\in \mathcal{K}$ for $\varphi\in
C(M)$, where $\mathcal{K}$ is the ideal of compact operators on $\mathcal{H}$.
In this note we consider two notions of span for essentially normal projections
$P$ and $Q$, and investigate when they are also essentially normal. First, we
show the representation theorem for two projections, and relate these results
to Arveson's conjecture for the closure of homogenous polynomial ideals on the
Drury-Arveson space. Finally, we consider the relation between the relative
position of two essentially normal projections and the $K$ homology elements
defined for them.
