5
$\begingroup$

Let $A$ be a bounded operator on a Hilbert space $H$ with two invariant subspaces $M$ and $N$ s.t. $N \subset M$, dim$(M \cap N^{\perp})> 1$, and have no invariant subspaces between $N$ and $M$. Then, show that, there exists an operator $B$ on $H$ which has no proper invariant subspace.

All I want a hint for constructing $B$ with the help of $A$ and given conditions, even a little hint will be appreciated. Thanks in advance.

$\endgroup$
0
$\begingroup$

The hint is to think about the invariant subspaces of $A$ the lie in $M\cap N^\perp$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.