Moore–Penrose inverse of an operator ($A^{†}$)

For a bounded operator $A:X\to Y$ between Hilbert spaces, the Moore–Penrose inverse maps admissible data to the unique minimum-norm least-squares solution. If $({\sigma }_j,u_j,v_j)$ is a singular system of a compact operator, thenon the data for which this series converges.

More precisely, its domain is $\mathrm{ran}A\oplus (\mathrm{ran}A{)}^{\perp }$. It vanishes on $(\mathrm{ran}A{)}^{\perp }$ and takes values in $(\ker A{)}^{\perp }$. An infinite-rank compact operator on infinite-dimensional spaces has singular values approaching zero, so its inverse on its range is generally unbounded; Picard criterion specifies admissible data.

A minimum-norm least-squares solution minimizes $\Vert Ax-y\Vert$ and, among all such minimizers, minimizes $\Vert x\Vert$. It lies in $(\ker A{)}^{\perp }$ and equals $A^{†}y$ when the Moore–Penrose inverse of an operator is defined at $y$.