Bounded Linear and Conjugate-Linear Maps on a Dense Subspace of a Complex Hilbert Space Extend Uniquely

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ and $K$ be complex Hilbert spaces, let $D\subseteq H$ be a linear subspace that is dense in $H$, and let $C\ge0$ be real. Continuity of a map $H\to K$ is that of Continuous Map Between Metric Spaces for the metrics given by the norms.

1. (Linear maps)¶ Let $T:D\to K$ be a linear map with $\lVert T\xi\rVert\le C\lVert\xi\rVert$ for every $\xi\in D$. Then there is exactly one $\widetilde{T}\in\mathcal{L}(H,K)$ with $\widetilde{T}\xi=T\xi$ for every $\xi\in D$. It satisfies $\lVert\widetilde{T}\rVert_{\mathrm{op}}\le C$; and if $\lVert T\xi\rVert=\lVert\xi\rVert$ for every $\xi\in D$, then $\lVert\widetilde{T}\xi\rVert=\lVert\xi\rVert$ for every $\xi\in H$.

2. (Conjugate-linear maps)¶ Let $S:D\to K$ be a map such that $S(\xi+\eta)=S\xi+S\eta$, $S(c\xi)=\overline{c}\,S\xi$ and $\lVert S\xi\rVert\le C\lVert\xi\rVert$ for all $\xi,\eta\in D$ and $c\in\mathbb{C}$. Then there is exactly one continuous map $\widetilde{S}:H\to K$ with $\widetilde{S}\xi=S\xi$ for every $\xi\in D$. It satisfies $\widetilde{S}(\xi+\eta)=\widetilde{S}\xi+\widetilde{S}\eta$, $\widetilde{S}(c\xi)=\overline{c}\,\widetilde{S}\xi$ and $\lVert\widetilde{S}\xi\rVert\le C\lVert\xi\rVert$ for all $\xi,\eta\in H$ and $c\in\mathbb{C}$; and if $\lVert S\xi\rVert=\lVert\xi\rVert$ for every $\xi\in D$, then $\lVert\widetilde{S}\xi\rVert=\lVert\xi\rVert$ for every $\xi\in H$.

3. (Equality of operators)¶ Let $A,B\in\mathcal{L}(H,K)$ and let $E\subseteq K$ be a linear subspace dense in $K$. If $\langle\eta,A\xi\rangle=\langle\eta,B\xi\rangle$ for all $\xi\in D$ and $\eta\in E$, then $A=B$. In particular $A=B$ whenever $A\xi=B\xi$ for every $\xi\in D$.