Adjacent cross-sections of the commutant of Hilbert space operators

Applying the techniques resulting the existence of almost invariant half-spaces, similarity models $\text{wh}T$ can be given for upper triangular operator-matrices $T=\left[\begin{array}{cc}A& C\\ 0& B\end{array}\right]$. The model $\text{wh}T$ is also an operator-matrix, containing two diagonal operators in the general case \cite{ker25}, and the unilateral shift $S$ together with a diagonal operator in the particular case when $A$ is similar to $S$ \cite{ker26}. Well-chosen compressions of operators in the commutant $\{\text{wh}T{\}}^{\mathrm{\prime }}$ form a linear manifold $\text{wh}\text{Ł}$ satisfying the condition that every $\text{wh}X\in \text{wh}\text{Ł}$ is transformed into $\text{wh}Y$ with $\text{rank}\text{wh}Y\le 2$ by a canonical mapping. Furthermore, a cyclcity property of $\{T{\}}^{\mathrm{\prime }}$ yields transitivity of $\text{wh}\text{Ł}$. In \cite{ker25} and \cite{ker26} the 3-dimensional cross-sections of $\text{wh}\text{Ł}$ have been investigated characterizing the canonical bases occurring in the corresponding subspaces of the matrix-algebra $M_3[\text{C}]$. In this paper new conditions are provided by studying matching of adjacent cross-sections.