It is clear, that \(\Delta\) maps \(\mathcal{P}_l\) to \(\mathcal{P}_{l-2}\). To show subjectivity, we use the inner product defined in (0x6927d77b) . Since \(\ran \Delta^{\perp } = \ker \Delta^*\), and multiplication by \(x^2\) is injective, it follows that \(\Delta\) maps \(\mathcal{P}_l\) onto \(\mathcal{P}_{l-2}\).
\[
\DeclareMathOperator{\ker}{ker}
\DeclareMathOperator{\ran}{ran}
\]