Unitary Operator

Let $H$ together with $\langle\cdot,\cdot\rangle$ be a \reftext{def:complex-hilbert-space-2026a}{complex Hilbert space}, with \reftext{def:inner-product-norm-2026a}{induced norm} $\lVert\cdot\rVert$, which is a \reftext{def:complex-normed-space-2026a}{norm} on $H$ by claim 2 of \ref{lem:inner-product-norm-is-norm-2026a}. Let $T$ be a \reftext{def:bounded-linear-operator-2026a}{bounded linear operator} on $H$.

The operator $T$ is \textbf{unitary} if it is surjective, that is, every $v\in H$ satisfies $v=T(u)$ for some $u\in H$, and if