  All the spaces considered are over $\bbc$. $Z$ represents any Banach space,
$L(Z)$ the space of all the bounded operators on $Z$, and $H$ any Hilbert
space. We will prove that for any unital proper weakly closed subalgebra
(upwcsa) $R\subsetneq L(Z)$, the algebra $R^\p=\{A^* \st A\in R\}$ has
nontrivial invariant subspaces (ntinvss). This solves in \lb particular, the
three most famous long standing open problems in operator theory, (1) the
transitive algebra problem, (2) the hyperinvariant subspace problem, and (3)
the invariant subspace problem. The transitive algebra problem was raised by R.
V. Kadison in 1955 and it asks if every upwcsa of $L(H)$ has ntinvss. This
proves also a conjecture of P. Enflo, that every operator on a reflexive space
has ntinvss and a conjecture of V. I. Lomonosov, that the adjoint of any Banach
space operator has ntinvss.
