Liftings and extensions of operators in Brownian setting

ABSTRACT

We investigate the operators T on a Hilbert space $\mathcal{H}$ which have 2-isometric liftings S with the property $S^{\ast }S\mathcal{H}\subset \mathcal{H}$. We show that such liftings are closely related to some extensions of T, which have $2\times 2$ 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.

Keywords:

• Brownian unitary operator
• extension
• 2-isometric lifting
• A-contraction

2010 Mathematics Subject Classifications:

• 47A05
• 47A15
• 47A20
• 47A63

Acknowledgments

The author was supported by a project financed by Lucian Blaga University of Sibiu and Hasso Plattner Foundation research Grants LBUS-IRG-2019-05. I would like to thank the referees for a careful reading of the manuscript and for useful suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author was supported by a project financed by Lucian Blaga University of Sibiu and Hasso Plattner Foundation research Grants LBUS-IRG-2019-05.