Left and right inverses in infinite dimensional spaces

Question

My reference text, in a discussion of a unitary linear operator $U$, states that: "In finite-dimensional spaces, the existence of a left inverse, $U^\dagger$, ($U^\dagger U=I$) implies that $U^\dagger$ is also a right inverse ($UU^\dagger = I$). Thus our definition need make no mention of left and right in finite dimensional spaces." Why is this limited to finite dimensional spaces? Is there a counter-example in an infinite dimensional space?

Answer 1

The canonical example is left and right shifts in sequences spaces. In the vector space of real sequences $\Bbb{R}^\Bbb{N}$, the maps \begin{align*} R &: \Bbb{R}^\Bbb{N} \to \Bbb{R}^\Bbb{N} : (x_1, x_2, x_3, \ldots) \mapsto(0,x_1,x_2,\ldots) \\ L &: \Bbb{R}^\Bbb{N} \to \Bbb{R}^\Bbb{N} : (x_1, x_2, x_3, \ldots) \mapsto(x_2, x_3, x_4, \ldots) \end{align*} satisfy $LR = I$, but $RL \neq I$. Moreover, if we restrict ourselves to $\ell^2$, with the usual inner product, $R = L^\dagger$. Note that $R$ is an isometry, but $L$ is not.