Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Nehari and Carathéodory-Fejér type extension results for operator-valued functions on groups


Author: Mihály Bakonyi
Journal: Proc. Amer. Math. Soc. 131 (2003), 3517-3525
MSC (2000): Primary 43A17, 47A57, 43A35, 47A20
DOI: https://doi.org/10.1090/S0002-9939-03-06897-7
Published electronically: February 20, 2003
MathSciNet review: 1991764
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $G$ be a compact abelian group having the property that its character group $\Gamma $ is totally ordered by a semigroup $P$. We prove that every operator-valued function $k$ on $G$ of the form $k(x)=\sum\limits_{\gamma \in (-P)}\gamma (x)k_{\gamma }$, such that the Hankel operator $H_k$ is bounded, has an essentially bounded extension $K$ with $\vert\vert K\vert\vert _{\infty }=\vert\vert H_k\vert\vert$. The proof is based on Arveson's Extension Theorem for completely positive functions on $C^*$-algebras. Among the corollaries we have a Carathéodory-Fejér type result for analytic operator-valued functions defined on such groups.


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

  • 1. V.M. Adamjam, D.Z. Arov, and M.G. Krein, Infinite Hankel matrices and generalized Carathéodory-Fejér and Riesz problems, Functional Anal. Appl., Vol. 2(1968), 1-18. MR 38:2591
  • 2. W.B. Arveson, Subalgebras of $C^*$-algebras, Acta Math., 123(1969), 141-224. MR 40:6274
  • 3. W.B. Arveson, Interpolation problems in nest algebras, J. Funct. Anal., Vol. 3(1975), 208-233. MR 52:3979
  • 4. M. Bakonyi, The extension of positive definite operator-valued functions defined on a symmetric interval of an ordered group, Proc. Amer. Math. Soc., Vol. 130, No. 5(2002), 1401-1406.
  • 5. M. Bakonyi and D. Timotin, The intertwining lifting theorem for ordered groups, to appear in J. Functional Anal.
  • 6. R. Bruzual and M. Dominguez, Extensions of operator valued positive definite functions and commutant lifting on ordered groups, J. Functional Anal., Vol. 185, No. 2(2001), 456-473. MR 2002g:43005
  • 7. C. Carathéodory and L. Fejér, Über den Zusammenhang der Extremen von harmonischen Funktionen mit ihren Koeffizienten und Über den Picard-Landauschen Satz, Rend. Circ. Mat. Palermo, Vol. 32(1911), 193-217.
  • 8. M. Dominguez, Interpolation and prediction problems for connected abelian groups, Integral Equations Operator Theory, Vol. 40, No. 2(2001), 212-230. MR 2002c:47030
  • 9. C. Foias and A.E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Birkhäuser Verlag, Basel-Boston-Berlin, 1990. MR 92k:47033
  • 10. H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables I, Acta Mathematica, Vol. 99(1958), 165-202. MR 20:4155
  • 11. M.G. Krein, Sur le probléme de prolongement des functions hermitiniennes positives et continues, Dokl. Akad. Nauk. SSSR, Vol. 26(1940), 17-22. MR 2:361h
  • 12. Z. Nehari, On bounded bilinear forms, Annals of Math., Vol. 65(1957), 153-162. MR 18:633f
  • 13. V. Paulsen, Completely Bounded Maps and Dilations, Pitman Research Notes in Mathematics, Vol. 146, New York, 1986. MR 88h:46111
  • 14. V.I. Paulsen, S.C. Power, and R.G. Smith, Schur products and matrix completions, J. Funct. Anal., Vol. 85(1989), 151-178. MR 90j:46051
  • 15. L. Rodman, I. Spitkovsky, and H.J. Woerdeman, Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions, Trans. Amer. Math. Soc., Vol. 350, No. 6(1998), 2185-2227. MR 98h:47023
  • 16. L. Rodman, I. Spitkovsky, and H.J. Woerdeman, Contractive extension problems for matrix valued almost periodic functions of several variables, J. Operator Theory, Vol. 47(2002), 3-35.
  • 17. L. Rodman, I. Spitkovsky, and H.J. Woerdeman, Multiblock problems for almost periodic matrix functions of several variables, New York Journal of Math., Vol. 7(2001), 117-148. MR 2002h:42013
  • 18. L. Rodman, I. Spitkovsky, and H.J. Woerdeman, Abstract band method via factorization, positive and band extensions of multivariable almost periodic matrix functions, and spectral estimation, to appear in the Memoirs of the AMS.
  • 19. W. Rudin, Fourier Analysis on Groups, Interscience Publishers, New York, 1962. MR 27:2808
  • 20. Z. Sasvári,
    Positive Definite and Definitizable Functions,
    Akademie Verlag, Berlin, 1994. MR 95c:43005
  • 21. I. Spitkovsky and H.J. Woerdeman, The Carathéodory-Fejér problem for almost periodic functions, J. Funct. Anal., Vol. 115(1993), 281-293. MR 94f:47020

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 43A17, 47A57, 43A35, 47A20

Retrieve articles in all journals with MSC (2000): 43A17, 47A57, 43A35, 47A20


Additional Information

Mihály Bakonyi
Affiliation: Department of Mathematics, Georgia State University, Atlanta, Georgia 30303-3083
Email: mbakonyi@cs.gsu.edu

DOI: https://doi.org/10.1090/S0002-9939-03-06897-7
Received by editor(s): March 6, 2002
Received by editor(s) in revised form: June 16, 2002
Published electronically: February 20, 2003
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society