 Moore--Penrose inverse of a Hilbert-space operator

For a bounded operator $A:X\to Y$ between Hilbert spaces, $A^{†}$ has domain $\mathcal{R}(A)\oplus \mathcal{R}(A{)}^{\perp }$, maps into $\mathcal{N}(A{)}^{\perp }$, and inverts the restriction of $A$ to $\mathcal{N}(A{)}^{\perp }$. It is bounded exactly when $\mathcal{R}(A)$ is closed.