Transitive subspaces of 3× 3 complex matrices

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 T on a complex Hilbert space H H has a diagonal operator inside itself; see [9]. Applying this result to a block-triangular operator T(H₁ ⊕ H₂) T ∈ L ( H 1 ⊕ H 2 ) , it can be proved that a translate of T T is similar to an operator T(H⁽⁴⁾) T ^ ∈ L ( H ( 4 ) ) with two diagonal entries D D , D* D ∗ and two entries F F , F * F ∗ of rank $$1$$ 1 . Given any operator Q= [Qi,j]₄ Q = [ Q i , j ] 4 in the commutant \' { T ^ } ′ of T T ^ , the operator entry Q4,1</jats:tex-m