Necessary conditions for existence of ${\Gamma }_n$-contractions and examples of ${\Gamma }_3$-contractions

Abstract

The fundamental result of B. Sz. Nazy states that every contraction has a coisometric extension and a unitary dilation. The isometric dilation of a contraction on a Hilbert space motivated whether this theory can be extended sensibly to families of operators. It is natural to ask whether this idea can be generalized, where the contraction $T$ is substituted by a commuting $n$-tuples of operators $(S_1,\cdots ,S_n)$ acting on some Hilbert space having ${\Gamma }_n$ as a spectral set. We derive the necessary conditions for the existence of a ${\Gamma }_n$-isometric dilation for ${\Gamma }_n$-contractions. Also we discuss an example of a ${\Gamma }_3$-contraction $(S_1,S_2,S_3)$ acting on some Hilbert space $\mathcal{H},$ which has a ${\Gamma }_3$-isometric dilation, but it fails to satisfy the following condition: $$E_1^{\ast }{E}_1-E_1{E}_1^{\ast }=E_2^{\ast }{E}_2-E_2{E}_2^{\ast },$$ where $E_1$ and $E_2$ are the fundamental operators of $(S_1,S_2,S_3),$ $(S_1,S_2)$ is a pair of commuting contractions and $S_3$ is a partial isometry. Thus, the set of sufficient conditions for the existence of a ${\Gamma }_3$-isometric dilation breaks down, in general, to be necessary, even when the ${\Gamma }_3$-contraction $(S_1,S_2,S_3)$ has the special structure as described above.

12 pages