Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-2010-05268-7
Published electronically: August 31, 2010
MathSciNet review: 2719693
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. J. R. Archer, Positivity and the existence of unitary dilations of commuting contractions, Oper. Theory Adv. Appl. 171 (2006), 17-35. MR 2308555 (2008f:47011)
  • 2. E. Artin, Über die Zerlegung definiter Funktionen in Quadrate, Hamb. Abh. 5 (1927), 100-115; The collected papers of Emil Artin, 273-288, Addison-Wesley, 1965. MR 0671416 (83j:01083)
  • 3. M. Bakonyi, The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group, Proc. Amer. Math. Soc. 130 (2002), 1401-1406. MR 1879963 (2003e:43005)
  • 4. M. Bakonyi, D. Timotin, Extensions of positive definite functions on free groups, J. Funct. Anal. 246 (2007), 31-49. MR 2316876 (2008g:43009)
  • 5. C. Berg, J. P. R. Christensen, P. Ressel, Harmonic Analysis on Semigroups, Springer, Berlin, 1984. MR 747302 (86b:43001)
  • 6. T. M. Bisgaard, The two-sided complex moment problem, Ark. Mat. 27 (1989), 23-28. MR 1004719 (91b:44015)
  • 7. T. M. Bisgaard, Extensions of Hamburger's theorem, Semigroup Forum 57 (1998), 397-429. MR 1640879 (99k:44015)
  • 8. T. M. Bisgaard, On perfect semigroups, Acta Math. Hungar. 79 (1998), 269-294. MR 1619811 (99d:43006)
  • 9. T. M. Bisgaard, If $ S\times T$ is semiperfect, is $ S$ or $ T$ perfect? Hokkaido Math. J. 29 (2000), 523-529. MR 1795489 (2001i:43010)
  • 10. T. M. Bisgaard, Characterization of moment multisequences by a variation of positive definiteness, Collect. Math. 52 (2001), 205-218. MR 1885219 (2002k:42017)
  • 11. T.M. Bisgaard, On the relation between the scalar moment problem and the matrix moment problem on $ *$-semigroups, Semigroup Forum 68 (2004), 25-46. MR 2027606 (2005b:43007)
  • 12. T. M. Bisgaard, P. Ressel, Unique disintegration of arbitrary positive definite functions on $ *$-divisible semigroups, Math. Z. 200 (1989), 511-525. MR 987584 (90b:43007)
  • 13. J. Bochnak, M. Coste, M.-F. Roy, Géometrie algébrique réelle, Ergeb. Math. Grenzgeb., vol. 12, Springer-Verlag, Berlin and New York, 1987. MR 949442 (90b:14030)
  • 14. R. Bruzual, M. Domínguez, Extension of locally defined indefinite functions on ordered groups, Integr. Equ. Oper. Theory 50 (2004), 57-81. MR 2091054 (2005g:47068)
  • 15. R. B. Burckel, Weakly Almost Periodic Functions on Semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963 (41:8562)
  • 16. A.P. Calderón and R. Pepinsky, On the phase of Fourier coefficients for positive real periodic functions, in Computing Methods and the Phase Problem in X-ray Crystal Analysis, Department of Physics, Pennsylvania State College, State College, Pa. (1952), 339-348.
  • 17. D. Cichoń, J. Stochel, F. H. Szafraniec, Incomplete Riesz-Haviland criterion.
  • 18. A. H. Clifford, G. B. Preston, The algebraic theory of semigroups. Vol. I.Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1961. MR 0132791 (24:A2627)
  • 19. R. E. Curto, L. A. Fialkow, An analogue of the Riesz-Haviland theorem for the truncated moment problem, J. Funct. Anal. 255 (2008), 2709-2731. MR 2464189 (2009i:47039)
  • 20. M. A. Dritschel, On factorization of trigonometric polynomials, Integr. Equ. Oper. Theory 49 (2004), 11-42. MR 2057766 (2005d:47034)
  • 21. B. Fuglede, The multidimensional moment problem, Expo. Math. 1 (1983), 47-65. MR 693807 (85g:44010)
  • 22. K. Furuta, N. Sakakibara, Radon perfectness of conelike $ *$-semigroups in $ Q^{(\infty)}$, Acta Math. Hungar. 91 (2001), 1-8. MR 1912357 (2003d:22004)
  • 23. J.-P. Gabardo, Extension of positive-defnite distributions and maximum entropy, Memoirs of the Amer. Math. Soc. 489, Amer. Math. Soc., Providence, 1993. MR 1139457 (93g:42005)
  • 24. J.-P. Gabardo, Tight frames of polynomials and the truncated trigonometric moment problem, J. Fourier Anal. Appl. 1 (1995), 249-279. MR 1353540 (96m:42013)
  • 25. J.-P. Gabardo, Trigonometric moment problems for arbitrary finite subsets of $ \boldsymbol Z^n$, Trans. Amer. Math. Soc. 350 (1998), 4473-4498. MR 1443194 (99a:42005)
  • 26. J.-P. Gabardo, Truncated trigonometric moment problems and determinate measures, J. Math. Anal. Appl. 239 (1999), 349-370. MR 1723065 (2000i:47028)
  • 27. J. S. Geronimo, H. J. Woerdeman, Positive extensions, Fejér-Riesz factorization and autoregressive filters in two variables, Annals of Mathematics 160 (2004), 839-906. MR 2144970 (2006b:42036)
  • 28. E. K. Haviland, On the momentum problem for distributions in more than one dimension, Amer. J. Math. 57 (1935), 562-568. MR 1507095
  • 29. E. K. Haviland, On the momentum problem for distributions in more than one dimension, II, Amer. J. Math. 58 (1936), 164-168. MR 1507139
  • 30. E. Hewitt, K. A. Ross, Abstract Harmonic Analysis, Vol. II, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer-Verlag, New York-Berlin, 1970. MR 0262773 (41:7378)
  • 31. E. Hewitt, H. S. Zuckerman, The $ l_1$-algebra of a commutative semigroup, Trans. Amer. Math. Soc. 83 (1956), 70-97. MR 0081908 (18:465b)
  • 32. T. W. Hungerford, Algebra, Graduate Texts in Mathematics 73, Springer-Verlag, New York, 1974. MR 600654 (82a:00006)
  • 33. Z. J. Jabłoński, J. Stochel, F. H. Szafraniec, Unitary propagation of operator data, Proceedings of the Edinburgh Mathematical Society 50 (2007), 689-699. MR 2360524 (2008h:47022)
  • 34. Y. Kilpi, Über das komplexe Momentenproblem, Ann. Acad. Sci. Fenn. Ser.A. I. 236 (1957), 32 pp. MR 0094660 (20:1172)
  • 35. M.G. Krein, Sur le problème du prolongement des fonctions hermitiennes positives et continues, Dokl. Akad. Nauk SSSR 26 (1940), 17-22 (Russian). MR 0004333 (2:361h)
  • 36. R. J. Lindahl, P. H. Maserick, Positive-definite functions on involution semigroups, Duke Math. J. 38 (1971), 771-782. MR 0291826 (45:916)
  • 37. P. H. Maserick, Spectral theory of operator-valued transformations, J. Math. Anal. Appl. 41 (1973), 497-507. MR 0343084 (49:7828)
  • 38. W. Mlak, Dilations of Hilbert space operators (General theory), Dissertationes Math. 153 (1978), 1-65. MR 496046 (81j:47008)
  • 39. Y. Nakamura, N. Sakakibara, Perfectness of certain subsemigroups of a perfect semigroup, Math. Ann. 287 (1990), 213-220. MR 1054564 (92b:43008)
  • 40. K. Nishio, N. Sakakibara, Perfectness of conelike $ *$-semigroups in $ \mathbf Q\sp k$, Math. Nachr. 216 (2000), 155-167. MR 1774907 (2001g:43005)
  • 41. M. Putinar, F.-H. Vasilescu, Solving moment problems by dimensional extension, Ann. of Math. 149 (1999), 1087-1107. MR 1709313 (2001c:47023b)
  • 42. B. Reznick, Uniform denominators in Hilbert's seventeenth problem, Math. Z. 220 (1995), 75-97. MR 1347159 (96e:11056)
  • 43. M. Riesz, Sur le probléme des moments, Troisième Note, Arkiv für Matematik, Astronomi och Fysik 17 (1923), 1-52.
  • 44. W. Rudin, The extension problem for positive-definite functions, Illinois J. Math. 7 (1963), 532-539. MR 0151796 (27:1779)
  • 45. W. Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062 (51:1315)
  • 46. L. A. Sakhnovich, Interpolation Theory and its Applications, Kluwer Academic Publishers, Dordrecht, 1997. MR 1631843 (99j:47016)
  • 47. K. Schmüdgen, The $ K$-moment problem for compact semi-algebraic sets, Math. Ann. 289 (1991), 203-206. MR 1092173 (92b:44011)
  • 48. J. A. Shohat, J. D. Tamarkin, The problem of moments, Math. Surveys, vol II, Amer. Math. Soc., Providence, Rhode Island, 1943. MR 0008438 (5:5c)
  • 49. J. Stochel, A note on general operator dilations over $ *$-semigroups, Bull. Acad. Polon. Sci. Sér. Sci. Math. 28 (1980), 149-153. MR 620351 (83m:47008)
  • 50. J. Stochel, Moment functions on real algebraic sets, Ark. Mat. 30 (1992), 133-148. MR 1171099 (93d:47032)
  • 51. J. Stochel and F. H. Szafraniec, On normal extensions of unbounded operators. I, J. Operator Theory 14 (1985), 31-55. MR 789376 (87d:47034)
  • 52. J. Stochel and F. H. Szafraniec, On normal extensions of unbounded operators. II, Acta Sci. Math. $ ($Szeged$ )$ 53 (1989), 153-177. MR 1018684 (91i:47032)
  • 53. J. Stochel, F. H. Szafraniec, On normal extensions of unbounded operators. III. Spectral properties, Publ. RIMS, Kyoto Univ. 25 (1989), 105-139. MR 999353 (91i:47033)
  • 54. J. Stochel, F. H. Szafraniec, The complex moment problem and subnormality: A polar decomposition approach, J. Funct. Anal. 159 (1998), 432-491. MR 1658092 (2001c:47023a)
  • 55. J. Stochel, F. H. Szafraniec, Unitary dilation of several contractions, Operator Theory: Adv. Appl. 127, Birkhäuser, Basel 2001, 585-598. MR 1902903 (2003d:47008)
  • 56. J. Stochel, F. H. Szafraniec, Domination of unbounded operators and commutativity, J. Math. Soc. Japan 55 (2003), 405-437. MR 1961294 (2003m:47033)
  • 57. F. H. Szafraniec, A general dilation theorem, Bull. Acad. Polon. Sci., Sér. Sci. Math. Astr. et Phys. 25(1977), 263-267. MR 0445309 (56:3651)
  • 58. F. H. Szafraniec, Boundedness of the shift operator related to positive definite forms: An application to moment problems, Ark. Mat. 19 (1981), 251-259. MR 650498 (84b:44015)
  • 59. F. H. Szafraniec, Dilations of linear and nonlinear operator maps, Functions, series, operators (Budapest, 1980), 1165-1169, Colloq. Math. Soc. János Bolyai, 35, North-Holland, Amsterdam, 1983. MR 751074
  • 60. F. H. Szafraniec, Sur les fonctions admettant une extension de type positif, C. R. Acad. Sci. Paris 292 (1981), Sér. I, 431-432. MR 611409 (82c:46075)
  • 61. B. Sz.-Nagy, Extensions of linear transformations in Hilbert space which extend beyond this space, Appendix to F. Riesz, B. Sz.-Nagy, Functional Analysis, Ungar, New York, 1960.
  • 62. B. Sz.-Nagy and C. Foiaş, Harmonic analysis of operators on Hilbert space, Akadémiai Kiadó, Budapest and North-Holland, Amsterdam, London, 1970. MR 0275190 (43:947)
  • 63. F.-H. Vasilescu, Existence of unitary dilations as a moment problem, Oper. Theory Adv. Appl. 143 (2003), 333-344. MR 2019357 (2005h:47018)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 43A35, 44A60, 47A20, 47B20

Retrieve articles in all journals with MSC (2010): 43A35, 44A60, 47A20, 47B20


Additional Information

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: https://doi.org/10.1090/S0002-9947-2010-05268-7
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.

American Mathematical Society