Proposition 3

— Let E and F be two hilbertian spaces and $u \in \mathscr{L}(E;F)$. The following conditions are equivalent :

(i) $u$ is a topological vector space isomorphism, with an inverse equal to $u^*$; (ii) $u$ is surjective and $u^*u = 1_E$; (iii) $u$ is injective and $uu^* = 1_F$; (iv) $u$ is an isomorphism of normed spaces; (v) $u$ is a hilbertian space isomorphism.

Condition (1) means that we have $u^*u = 1_E$ and $uu^* = 1_F$. Hence the equivalence of (i), (ii) and (iii) follows from E, II, § 3, No. 8, prop. 8. We have already seen the equivalence of (iv) and (v) (V, p. 5). Finally, the relation $u^*u = 1_E$ is equivalent to

$^1$ See P. A. M. Dirac, Quantum Mechanics, Oxford University Press, New York, 1935.

$\langle x|u^*u(y)\rangle = \langle x|y\rangle$, that is, to $\langle u(x)|u(y)\rangle = \langle x|y\rangle$ for all $x, y$ in $E$, and evidently implies that $u$ is injective; this proves the equivalence of (ii) and (v).

An automorphism of the hilbertian space $E$ is also called a unitary operator, that is, an operator $u \in \mathscr{L}(E)$ satisfying $uu^* = u^*u = 1_E$.