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Partially isometric dilations of noncommuting $N$-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: 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

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