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For a bounded operator $A:X\to Y$ between Hilbert spaces, the Moore--Penrose inverse of a Hilbert-space operator is defined onand has range $\mathcal{R}(A^{†})=\mathcal{N}(A{)}^{\perp }$. Its Penrose equations areEquivalently, $A^{†}A=P_{\mathcal{N}(A{)}^{\perp }}$ and $A{A}^{†}$ is the restriction of $P_{\overline{\mathcal{R}(A)}}$ to $\mathcal{D}(A^{†})$.

The Reverse-order law for the Moore--Penrose inverse is false in general. TakeThenA sufficient condition isso that $A$ has full column rank and $B$ has full row rank. Indeed $A^{†}A=I_n$ and $B{B}^{†}=I_n$; these identities make $B^{†}{A}^{†}$ satisfy all four Penrose equations for $AB$.