Has anyone an example of a linear contin. operator T of a Hilbert space on C with a residual spectral value c for which T-c.Id has unbounded inverse? Let H be a complex Hilbert space (with infinite dimension - best countable, in which case it will be isomorphic to $\ell^2$ and many other typical H. spaces), T a linear continuous operator on H having a non empty residual spectrum $S_r$. I would like to receive an example where there is at least one number c in $S_r$ for which T-c.Id (Id=identity map) has a non continuous (equiv. unbounded) inverse. Thanks
 A: Suppose H is a Hilbert space of infinite countable Hilbert dimension, $\displaystyle(e_n)_{n \scriptstyle \in \mathbb N}$ a Hilbert base of H, T and U are continuous injective but not surjective operators of H with T(H) closed and U(H) dense (in H). For example, we can use T = the right shift on said base, and U = S-c.I where S=T or S = the left shift on the same base, c is any complex nb. such that |c|=1 (especially c=1 or c=-1 are simplest cases) ... it is well known that c is in the continuous spectrum of S, showing that this choice for U is OK. Now consider the operator W=TU. W is continuous, injective and W(H) = T(U(H)). Since T is a topological vector spaces isomorphism (in fact even of Banach spaces in my example) H -> T(H), the fact that U(H) is dense in H and not= H carries over to: T(U(H)) is dense in T(H) and not= T(H). This implies that W(H) is 1° not dense in H (being contained in a proper closed subspace) 2° not closed in H (since it is not closed in T(H) ). This result can be interpreted like this: W has z=0 in its residual spectrum with the inverse of W-z.I (=W) not continuous
