Isometric dilation
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Given a commuting tuple of operators forming a row contraction there are two commonly used dilations in multivariable operator theory.
Firstly there is the minimal isometric dilation consisting of isometries with orthogonal ranges and hence it is a noncommuting tuple.
There is also a commuting dilation related with a standard commuting tuple on boson Fock space. This commuting dilation is the maximal commuting piece of the minimal isometric dilation.
