91Ó°ÊÓ

Let \(H\) be a Hilbert space and \(T: H \longrightarrow H\) a bijective bounded linear operator whose inverse is bounded. Show that \(\left(T^{*}\right)^{-1}\) exists and $$ \left(T^{*}\right)^{-1}=\left(T^{-1}\right)^{*} $$

Show that in an inner product space, \(x \perp y\) if and only if \(\|x+\alpha y\| \geq\|x\|\) for all scalars \(\alpha\).

Show that the annihilator \(M^{\perp}\) of a set \(M \neq \varnothing\) in an inner product space \(X\) is a closed subspace of \(X\).

Show that an isometric linear operator \(T: H \longrightarrow H\) which is not unitary maps the Hilbert space \(H\) onto a proper closed subspace of \(H\).