Adjacent cross-sections of the commutant of Hilbert space operators

Abstract

Applying the techniques resulting the existence of almost invariant half-spaces, similarity models $\wh T$ can be given for upper triangular operator-matrices $T= \left[\begin{matrix}A&C\\ 0&B\end{matrix}\right]$. The model $\wh T$ 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 $\{\wh T\}'$ form a linear manifold $\wh\L$ satisfying the condition that every $\wh X\in\wh \L$ is transformed into $\wh Y$ with $\rank\wh Y\le 2$ by a canonical mapping. Furthermore, a cyclcity property of $\{T\}'$ yields transitivity of $\wh\L$. In \cite{ker25} and \cite{ker26} the 3-dimensional cross-sections of $\wh\L$ have been investigated characterizing the canonical bases occurring in the corresponding subspaces of the matrix-algebra $M_3[\C]$. In this paper new conditions are provided by studying matching of adjacent cross-sections.