= Moore–Penrose inverse of an operator
{c}
{title2=$A^\dagger$}
= Moore–Penrose generalized inverse operator
{c}
{synonym}
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>, then
$$
A^\dagger y
=\sum_j\frac{\langle y,u_j\rangle}{\sigma_j}v_j
$$
on the data for which this series converges.
More precisely, its domain is $\operatorname{ran}A\oplus(\operatorname{ran}A)^\perp$. It vanishes on $(\operatorname{ran}A)^\perp$ 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.
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