It was a remarkable result of the last decades that every Banach space operator has an almost invariant half-space; see [1] and [17].Refining the technique used in [1], it has been shown quite recently that every operator \(T\) on a complex Hilbert space \(\mathcal{H}\) has a diagonal operator inside itself; see [9].Applying this result to a block-triangular operator \(T\in\mathcal{L}(\mathcal{H}_1 \oplus \mathcal{H}_2)\) ,it can be proved that a translate of \(T\) is similar to an operator \(\widehat{T}\in\mathcal{L}(\mathcal{H}^{(4)})\) with two diagonal entries \(D\) , \(D_{*} \) and two entries \(F\) , \(F _{*} \) of rank \(1\) .Given any operator \(Q= [Q_{i,j}]_4\) in the commutant \(\{\widehat{T}\}'\) of \(\widehat{T}\) , the operator entry \(Q_{4,1}\) intertwines \(D\) and \(D _{*} \) up to a transformation of rank at most \(2\) .The linear manifold of the operators \(Q_{4,1}\) is denoted by \(\mathcal{L}_{4,1}\) .The compressions of the transformations in \(\mathcal{L}_{4,1}\) to a \(3\) -dimensional subspace form a subspace \(\mathcal{L} _{*} \) of the matrix algebra \(M_3[\mathbb{C}]\) .Transitivity properties of \(\{ T\}'\) yield the transitivity of \(\mathcal{L} _{*} \) .Our aim is to characterize all transitive subspaces \(\mathcal{L}\) of \(M_3[\mathbb{C}]\) obeying the transformation law derived from the intertwining condition on \(\mathcal{L}_{4,1}\) .In that way we obtain sufficient conditions for the existence of proper hyperinvariant subspaces of \(T\) .
