Mathematical Foundations for Signal Processing, Communications, and Networking

Theorem 20.3.7. Let $\mathbb{A}$ : $\mathscr{H}$ → l² be a bounded linear operator. Assume that $\mathbb{A}$* $\mathbb{A}$ : $\mathscr{H}$→$\mathscr{H}$ is invertible on $\mathscr{H}$. Then, the operator $\mathbb{T}$⁺ : l² → $\mathscr{H}$ defined as $\mathbb{T}$⁺ ≜ ($\mathbb{A}$*$\mathbb{A}$)⁻¹$\mathbb{A}$* is a left-inverse of $\mathbb{A}$, i.e., $\mathbb{A}$⁺ $\mathbb{A}$ = $\mathbb{I}$_{$\mathscr{H}$}, where $\mathbb{I}$_{$\mathscr{H}$} is the identity operator on $\mathscr{H}$. Moreover, the general solution $\mathbb{T}$ of the equation $\mathbb{LA}$ = $\mathbb{I}$_{$\mathscr{H}$} is given by

$\mathbb{L}={\mathbb{A}}^{+}+\mathbb{M}\left({\mathbb{I}}_{l^2}-\mathbb{A}{\mathbb{A}}^{+}\right),$

where $\mathbb{M}$ : l² → $\mathscr{H}$ is an arbitrary bounded linear operator and ${\mathbb{I}}_{l^2}$ is the identity operator on l².

Applying this theorem to the operator $\mathbb{T}$ we see that all left-inverses of $\mathbb{T}$ can be written as

$\mathbb{L}={\mathbb{T}}^{+}+\mathbb{M}\left({\mathbb{I}}_{l^2}-\mathbb{T}{\mathbb{T}}^{+}\right),$

(20.51)

where $\mathbb{M}$ : l² → $\mathscr{H}$ is an arbitrary bounded linear operator and

${\mathbb{T}}^{+}={\left({\mathbb{T}}^{*}\mathbb{T}\right)}^{-1}{\mathbb{T}}^{*}.$

Now, using (20.48), we obtain the following important identity:

${\mathbb{T}}^{+}={\left({\mathbb{T}}^{*}\mathbb{T}\right)}^{-1}{\mathbb{T}}^{*}={\mathbb{S}}^{-1}{\mathbb{T}}^{*}={\tilde{\mathbb{T}}}^{*}.$

This ...