Liftings and extensions of operators in Brownian setting.

We investigate the operators T on a Hilbert space H which have 2-isometric liftings S with the property S ∗ S H ⊂ H . We show that such liftings are closely related to some extensions of T, which have 2 × 2 upper triangular block matrices having contractive entries. We describe these extensions by the corresponding 2-isometric liftings, and respectively as regular A-contractions for some positive operators A. Also, we show that some extensions have Brownian isometric (or unitary) liftings and, in particular, we refer to expansive operators with such liftings. [ABSTRACT FROM AUTHOR]