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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Extending positive definiteness


Authors: Dariusz Cichoń, Jan Stochel and Franciszek Hugon Szafraniec
Journal: Trans. Amer. Math. Soc. 363 (2011), 545-577
MSC (2010): Primary 43A35, 44A60; Secondary 47A20, 47B20
Published electronically: August 31, 2010
MathSciNet review: 2719693
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Abstract: The main result of this paper gives criteria for extendibility of mappings defined on symmetric subsets of $ *$-semigroups to positive definite ones. By specifying the mappings in question we obtain new solutions of relevant issues in harmonic analysis concerning truncations of some important multivariate moment problems, like complex, two-sided complex and multidimensional trigonometric moment problems. In addition, unbounded subnormality and existence of unitary power dilation of several contractions is treated as an application of our general scheme.


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Dariusz Cichoń
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland
Email: Dariusz.Cichon@im.uj.edu.pl

Jan Stochel
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland
Email: Jan.Stochel@im.uj.edu.pl

Franciszek Hugon Szafraniec
Affiliation: Instytut Matematyki, Uniwersytet Jagielloński, ul. Łojasiewicza 6, PL-30348 Kraków, Poland
Email: Franciszek.Szafraniec@im.uj.edu.pl

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05268-7
PII: S 0002-9947(2010)05268-7
Keywords: Positive definite mapping, completely positive mapping, completely f-positive mapping, complex moment problem, multidimensional trigonometric moment problem, truncated moment problems, sums of squares (SOS), the 17th Hilbert problem, subnormal operator, unitary power dilation for several contractions
Received by editor(s): December 22, 2008
Received by editor(s) in revised form: December 9, 2009
Published electronically: August 31, 2010
Additional Notes: This work was partially supported by KBN grant 2 P03A 037 024 and by MNiSzW grant N201 026 32/1350. The third author would like to acknowledge assistance of the EU Sixth Framework Programme for the Transfer of Knowledge “Operator theory methods for differential equations” (TODEQ) # MTKD-CT-2005-030042. A very early version of this paper was designated for Kreĭn’s anniversary volume. However, due to its growing capacity we have been exceeding all consecutive deadlines; let us thank Professor Vadim Adamyan for his patience in negotiating them.
Dedicated: To the memory of M.G. Kreĭn (1907-1989)
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.