extending bounded linear map

Question

Let $H_0$ be a Hilbert subspace of a Hilbert space $H$. Suppose $T_0 \in B(H_0)$. Can I extend $T_0$ uniquely to $T \in B(H)$? I know extension exists. Can it be unique?

Answer 1

It cannot be extended uniquely, not even in the finite dimensional case. For example, take $\mathbb{R}^3$ and a basis $\{e_1,e_2,e_3\}$ of $\mathbb{R}^3$. Let $T_0(e_1)=e_1$ in $H_0=span(e_1)$. Then $T_1(e_2)=e_3,\ T_1(e_3)=e_2$ and $T_2=I$ ($I$ the identity). Then $T_1,\ T_2$ are two different extensions of $T$.