Dilations and Commutant Lifting for Jointly Isometric Operators—A Geometric Approach

A tuple of commuting contractionsT=(T1, T2, …, Tn) is called a joint-isometry if ∑ T*jTj=I. We give a geometric proof that joint isometries have a regular unitary dilation and that its commutant lifts. We also show thatTis subnormal and that its minimal normal extension is also jointly isometric.