3 added 11 characters in body

Linear case

In the linear case, these identities are part of the definition of the Moore-Penrose pseudo-inverse, which exists and is unique. Given $A\in M_{p\times q}(\mathbb C)$, its MPpi is the matrix $A^\dagger\in M_{q\times p}(\mathbb C)$ that satisfies $$AA^\dagger A=A,\qquad A^\dagger AA^\dagger=A^\dagger,\qquad(AA^\dagger)^H=AA^\dagger,\qquad(A^\dagger A)^H=A^\dagger A,$$ where the superscript $H$ stands for the Hermitian adjoint.

If $A\in GL_n(\mathbb C)$, then $A^\dagger=A^{-1}$. But otherwise, $AA^\dagger$ and $A^\dagger A$ are only unitary projections.

Nonlinear case

The situation where $f=g$ is amazing: one looks at functions $h$ such that $h\circ h\neq {\rm id}$, whereas $h\circ h\circ h=h$. Then we have $h^{(2k)}=h^2$ and $h^{(2k-1)}=h$ for every $k\ge1$.

Such an $h$ can be obtained by the following construction, when we are given $f,g$ such that $fgf=f$, $gfg=g$ and at least one of $fg$ or $gf$ is not the identity. Just define $h=(f,g)$ h(x,y)=(f(x),g(y))$on the cartesian product. Application: take for$f$the backward shift on$\ell^p({\mathbb N})$and for$g$the forward shift. 2 added 557 characters in body; added 2 characters in body Linear case In the linear case, these identities are part of the definition of the Moore-Penrose pseudo-inverse, which exists and is unique. Given$A\in M_{p\times q}(\mathbb C)$, its MPpi is the matrix$A^\dagger\in M_{q\times p}(\mathbb C)$that satisfies $$AA^\dagger A=A,\qquad A^\dagger AA^\dagger=A^\dagger,\qquad(AA^\dagger)^H=AA^\dagger,\qquad(A^\dagger A)^H=A^\dagger A,$$ where the superscript$H$stands for the Hermitian adjoint. If$A\in GL_n(\mathbb C)$, then$A^\dagger=A^{-1}$. But otherwise,$AA^\dagger$and$A^\dagger A$are only unitary projections. Nonlinear case The situation where$f=g$is amazing: one looks at functions$h$such that$h\circ h\neq {\rm id}$, whereas$h\circ h\circ h=h$. Then we have$h^{(2k)}=h^2$and$h^{(2k-1)}=h$for every$k\ge1$. Such an$h$can be obtained by the following construction, when we are given$f,g$such that$fgf=f$,$gfg=g$and at least one of$fg$or$gf$is not the identity. Just define$h=(f,g)$on the cartesian product. Application: take for$f$the backward shift on$\ell^p({\mathbb N})$and for$g$the forward shift. 1 In the linear case, these identities are part of the definition of the Moore-Penrose pseudo-inverse, which exists and is unique. Given$A\in M_{p\times q}(\mathbb C)$, its MPpi is the matrix$A^\dagger\in M_{q\times p}(\mathbb C)$that satisfies $$AA^\dagger A=A,\qquad A^\dagger AA^\dagger=A^\dagger,\qquad(AA^\dagger)^H=AA^\dagger,\qquad(A^\dagger A)^H=A^\dagger A,$$ where the superscript$H$stands for the Hermitian adjoint. If$A\in GL_n(\mathbb C)$, then$A^\dagger=A^{-1}$. But otherwise,$AA^\dagger$and$A^\dagger A\$ are only unitary projections.