The following conjeture is stated here:
Every adjoint operator has a non-trivial closed invariant subspace.
Reference 11 where adjoint is supposedly defined can be found here. But I don't have access to the article.
So what's an "adjoint operator"?
Edit It is not "the adjoint of an operator" here because then the conjecture would be equivalent to:
Every operator has a non-trivial closed invariant subspace.
For a Banach space $X$ , the dual space is the Banach space $X^*$ of all bounded linear functionals on $X$. Let $\langle x,f\rangle$ be the value of $f\in X^*$ on $x\in X$. If $T$ is a bounded linear operator on $X$, then its adjoint is a bounded linear operator $T^*$ on $X^*$ which is defined by $\langle x,T^*f\rangle=\langle Tx,f\rangle$ for all $x\in X$ and $f\in X^*$. There exist Banach space $X$ such that not every bounded linear operator on $X^*$ is the adjoint of an operator on $X$. The question if every operator on $X^*$ which is an adjoint has a non-trivial closed invariant subspace is stil unsolved. The problem is open even for the infinite dimensional separable complex Hilbert space. It is known, for instance, that there exists a bounded linear operator $S$ on $l^1$ which does not have non-trivial closed invariant subspaces. Of course, $S$ is not an adjoint operator.