Brownian Type Extensions for a Class of m-Isometries

The class of m-isometries on a Hilbert space which admit Brownian unitary extensions is investigated. Brownian unitaries in this context for integers m =3, naturally appear as a generalization of the same concept defined by Agler-Stankus for 2-isometries. By contrast with the case m = 2, here just certain expansive m-isometries have Brownian unitary extensions, when m =3. In this context we describe the m-Brownian unitaries, as well as the operators which have such extensions, these being called sub-Brownian m-isometries. Also we characterize the operators which have sub-Brownian m-isometric liftings, obtained by the coupling of an isometry. We refer here, in particular, to the operators similar to compressions of sub-Brownian (m - 1)-isometries. As examples we de-scribe some weighted shifts with scalar weights, which are sub-Brownian m-isometries. Our description is given in terms of the polynomial which determine the weights of such an operator. As an application we prove that the operators with polynomial growth conditions of the powers have m-Brownian unitary dilations.