Partially isometric dilations of noncommuting -tuples of operators

Authors:
Michael T. Jury and David W. Kribs

Journal:
Proc. Amer. Math. Soc. **133** (2005), 213-222

MSC (2000):
Primary 47A20, 47A45

DOI:
https://doi.org/10.1090/S0002-9939-04-07547-1

Published electronically:
June 23, 2004

MathSciNet review:
2085172

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Abstract | References | Similar Articles | Additional Information

Abstract: Given a row contraction of operators on a Hilbert space and a family of projections on the space that stabilizes the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries that satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold decomposition for partial isometries to describe the models for these dilations, and we discuss how the basic properties of a dilation depend on the row contraction.

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Additional Information

**Michael T. Jury**

Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907

Email:
jury@math.purdue.edu

**David W. Kribs**

Affiliation:
Department of Mathematics and Statistics, University of Guelph, Guelph, Ontario, Canada N1G 2W1

Email:
dkribs@uoguelph.ca

DOI:
https://doi.org/10.1090/S0002-9939-04-07547-1

Keywords:
Hilbert space,
operator,
row contraction,
partial isometry,
minimal dilation,
directed graph

Received by editor(s):
June 13, 2003

Received by editor(s) in revised form:
September 29, 2003

Published electronically:
June 23, 2004

Communicated by:
Joseph A. Ball

Article copyright:
© Copyright 2004
American Mathematical Society