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A Wold-type decomposition for commuting isometric pairs

Author: Dan Popovici
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2303-2314
MSC (2000): Primary 47A13, 47A45
Published electronically: February 26, 2004
MathSciNet review: 2052406
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Abstract: We obtain a Wold-type decomposition theorem for an arbitrary pair of commuting isometries $V$ on a Hilbert space. More precisely, $V$ can be uniquely decomposed into the orthogonal sum between a bi-unitary, a shift-unitary, a unitary-shift and a weak bi-shift part, that is, a part $S=(S_1,S_2)$ that can be characterized by the condition that $S_1S_2, S_1\vert _{\bigcap_{n\ge 0}\ker S_2^*S_1^n}$ and $S_2\vert _{\bigcap_{n\ge 0}\ker S_1^*S_2^n}$ are shifts. Moreover, $S$ contains bi-shift and modified bi-shift maximal parts.

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Additional Information

Dan Popovici
Affiliation: Department of Mathematics, University of the West Timişoara, RO-300223 Timişoara, Bd. Vasile Pârvan nr. 4, Romania

Keywords: Wold-type decomposition, (dual) bi-isometry, (weak, modified) bi-shift, unitary extension
Received by editor(s): September 13, 2002
Received by editor(s) in revised form: April 21, 2003
Published electronically: February 26, 2004
Additional Notes: This work was supported by the EEC Research Training Network: “Analysis and Operators”, contract no. HPRN-CT-2000-00116
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2004 American Mathematical Society