On the solution of Kim, Pearcy and Shields problem and its consequences

We give a negative answer to a question due to Kim, Pearcy and Shields, namely, we prove that every nonscalar bounded linear operator on a Hilberts space $\mathcal{H}$ belongs to the class $\mathrm{\Delta }(\mathcal{H})$ defined in [56]. This gives an affirmative solution to the (hyper) invariant subspace problem in $\mathcal{H}$. Our method of proof is based on Berezin symbol technique in reproducing kernel Hilbert spaces.