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Unitary extensions of partial isometries
Author(s) -
Amoretti Nieves,
Domínguez Marisela
Publication year - 2011
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810211
Subject(s) - mathematics , multiplicative function , unitary state , combinatorics , order (exchange) , extension (predicate logic) , mathematical analysis , finance , political science , computer science , law , economics , programming language
If ( S ( n , m ) ) ( n , m ) ⩾ (0, 0) is a multiplicative family of partial isometries on \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\mathbb {Z}^{2}$\end{document} with the lexicographic order, then S (1, 0) and S (0, 1) commute in a certain weak sense. Let \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\widetilde{A}$\end{document} and $\widetilde{B}$ be commuting unitary extensions of S (1, 0) and S (0, 1) . We give a sufficient condition for \documentclass{article}\usepackage{amssymb}\pagestyle{empty}\begin{document}$\big (\widetilde{A}^{n} \widetilde{B}^{m}\big )_{(n,m) \in \mathbb {Z}^{2}}$\end{document} to be a unitary extension of the given family. Under this condition we present a description of the set of extensions. We also describe the set of all minimal commuting unitary extensions of any pair of partial isometries that commutes in the same weak sense that S (1, 0) and S (0, 1) commute. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim