Let $E$ an **infinite dimensional** complex Hilbert space and $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.


**Definition:** Let $T \in \mathcal{L}(E)$. The Moore-Penrose inverse of $T$, denoted by $T^{+}$, is defined as the unique linear extension of $(\bar{T})^{-1}$ in
 $$D(T^{+}) = \mathcal{R}(T)+\mathcal{R}(T)^{\perp},$$
with $\mathcal{N}(T^{+}) = \mathcal{R}(T)^{\perp}$ and $\bar{T}$ is the isomorphism
$$\bar{T}:=T|\_{{\mathcal{N}(T)}^{\perp}}: {\mathcal{N}(T)}^{\perp} \longrightarrow \mathcal{R}(T).$$
Moreover, $T^{+}$ is the unique solution of the four ''Moore-Penrose equations'':
$$TXT = T,\quad XTX = X,\quad XT = P\_{N{(T)^{\bot}}}\,\,\mbox{and}\,\,\quad TX = P\_{\overline{\mathcal{R}(T)}}{{|}\_{D(T^{+})}}.$$


Here $\mathcal{R}(T)$ and $\mathcal{N}(T)$ denote respectively the range and the nullspace of $T$. Also $P\_{F}$ denote the orthogonal projection onto $F$.



> 
> It is well know that $A^+$ is bounded iff $A$ has a closed range.
> 
> 
> * If $A$ is selfadjoint matrix, then
>  $$ A^{+}= \lim\_{t \to 0}(A^2+tI)^{-1} A.\;\;(1).$$
> * If $A$ is not selfadjoint, then $A^+=A^\*(AA^\*)^+$ (which is equal to $(A^\*A)^+A^\*$).
> 
> 
> Is the formula $(1)$ true only for matrix or even for Hilbert space operators? Let for example $S$ an operator on $\ell^2$ given by 
>  $$S(x\_1,x\_2,\cdots)=(x\_1,\tfrac{1}{2} x\_2,\tfrac{1}{3} x\_3,\cdots).$$
>  How can I find $S^+$?
> 
> 
> 



A formula related to the Moore-Penrose pseudo-inverse of Hilbert space operators
What may be tripping you up here is that $T^+$, as you have defined it, could be unbounded, if the range of $T$ is not closed. So the answer to



> 
> Is the formula (1) true only for matrix or even for Hilbert space operators?
> 
> 
> 


is ... sort of, depending on what you mean by taking a limit. I'm not sure there's any standard notion of convergence for unbounded operators. But you can make sense of (1) as a strong limit on the domain of $T^+$.


Your second question is easier: $S^+(x\_1, x\_2, x\_3, \ldots) = (x\_1, 2x\_2, 3x\_3, \ldots)$, with domain the set of sequences in $l^2$ for which that target sequence also belongs to $l^2$.


