Let H1 and H2 be Hilbert spaces..If T:H1-->H2 is a bounded linear transformation with M1 closed subspace of H1 and M2 closed subspace of H2 then show that T(M1) is contained in M2 if and only if T*(M2) is contained in M1..
Any help appreciated.
The simplest way to see that it is wrong is to think about what happens if and . Then the question is saying that if a subspace is invariant under then it is also invariant under . That is certainly not true. For example, in a 2-dimensional space, the subspace spanned by the second basis vector is invariant under the transformation with matrix , but not under its adjoint.