Properties of Unitary Operators

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:unitary-operator-2026a}{unitary operator} on $H$, and write $\lVert T\rVert_{\mathrm{op}}$ for its \reftext{def:operator-norm-2026a}{operator norm}, which exists and is unique by \ref{lem:operator-norm-existence-uniqueness-2026a}; as there, $\lVert\cdot\rVert$ without a subscript denotes the norm of a vector of $H$. Then the following hold.

\textbf{1. (Preservation of the norm)} For every $u\in H$,

\textbf{2. (Bijectivity)} $T$ is a bijection from $H$ onto $H$.

\textbf{3. (Operator norm)} $\lVert T\rVert_{\mathrm{op}}\le1$, the order being that of the \reftext{def:ordered-field-c54-2026b}{ordered field} of \reftext{def:real-numbers-c54-2026c}{real numbers}.