# Integral Representations of Bounded Hankel Forms Defined in Scattering Systems with a Multiparametric Evolution Group

## Abstract

The notion of Hankel and Toeplitz forms acting in algebraic scattering systems is introduced. Particular examples of such systems are the classical scattering structures on Hilbert spaces as well as the trigonometric polynomials under the shift operator. In this last example, the Hankel and Toeplitz forms acting in it coincide with the usual ones. Given two positive Toeplitz forms in a scattering system, the Hankel forms bounded with respect to the hilbertian seminomas they define, have Toeplitz extensions similarly bounded, as well as Fourier representations. These results still hold when the discrete 1-parametric evolution group of the system is replaced by a continuous or a *d*-parametric one. Several applications to harmonic analysis are obtained.

## Keywords

Matrix Measure Positive Form Fourier Representation Lift Property Scattering System## Preview

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## References

- [1]V.M. Adamjan,
*Non-degenerate unitary coupling of semiunitary operators*, Funk. An. Priloz,**7**: 4 (1973), 1–16 ( in Russian).Google Scholar - [2]V.M. Adamjan and D.Z. Arov,
*On unitary couplings of semiunitary operators*, Matern. Issledovanya,**1**: 2 (1966), 3–64 ( in Russian).Google Scholar - [3]R. Arocena and M. Cotlar,
*Dilations of generalized Toeplitz kernels and L**2**-weighted problems*, Lecture Notes in Math.,**908**, 1982, 169–188.CrossRefGoogle Scholar - [4]R. Arocena, M.Cotlar and C. Sadosky,
*Weighted inequalities in L**2**and lifting properties*, Math. Anal. & Appl., Adv. in Math. Suppl. Stud.,**7A**(1981), 95–128.Google Scholar - [5]R. Bruzual, Acta Cient. Venez., to appear.Google Scholar
- [6]M. Cotlar and C. Sadosky,
*On the Helson-Szegő theorem and a related class of modified Toeplitz kernels*, Proc. Symp. Pure Math. AMS,**25**: I (1979), 383–407.Google Scholar - [7]M. Cotlar and C. Sadosky,
*A lifting theorem for subordinated invariant kernels*, J. Funct. Anal.,**67**(1986), 345–359.CrossRefGoogle Scholar - [8]M. Cotlar and C. Sadosky,
*Lifting properties, Nehari theorem and Paley lacunary inequality*, Rev. Mat. Iberoamericana,**2**, 55–71.Google Scholar - [9]M. Cotlar and C. Sadosky,
*Toeplitz liftings of Hankel forms*, in “Function Space & Applications, Lund 1981”, (Eds.: J. Peetre, Y. Sagher & H. Wallin ), Lecture notes in Math., Springer-Verlag, New York, 1987.Google Scholar - M. Cotlar and C. Sadosky,
*Prolongements des formes de Hankel généralisées en**formes de Toeplitz*, CR. Acad. Sci. Paris A, # (1987).Google Scholar - [11]N. Dunford and J.T. Schwartz,
*Linear Operators*, Part I: General Theory, Interscience Publ., New York, 1958.Google Scholar - [12]H. Helson & D. Lowdenslager,
*Prediction theory and Fourier series in several variables*, Acta Math.,**99**(1958), 165–202.CrossRefGoogle Scholar