The following is Exercise 6.4.2 of Conway's Functional Analysis:
Suppose $T\in B(X)$ where $X$ is a Banach space. Prove that $M$ is invariant under $T$ if and only if $M^{\perp}$ is invariant under $T^*$. What does the map $M\to M^{\perp}$ of $Lat T$ into $Lat T^*$ do to the lattice operations? ($Lat T$ is the collection of all invariant subspaces for $T$.)
I'm confused by this exercise. I know that every closed subspace of a Banach space is not complemented. So how can I say for invariant subspace M in a infinite dimensional Banach space , there is a complemented subspace and then show that it's invariant under $T^*$?
And also how can I define the map of $Lat T$ into $Lat T^*$?
Please help me. Thanks in advance.