A von Neumann type inequality for certain domains in 𝐂ⁿ

Ambrozie, C.-G.; Timotin, D.

doi:10.1090/S0002-9939-02-06321-9

A von Neumann type inequality for certain domains in $\mathbf C^n$
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Proc. Amer. Math. Soc. 131 (2003), 859-869 Request permission

Abstract:

Strict contractions on a Hilbert space have a functional calculus with functions that are analytic in the unit disc of the complex plane; an estimate of the norm is then provided by von Neumann’s inequality. We consider functions that satisfy related inequalities with respect to multioperators connected to certain domains in ${\mathbf C}^n$; a representation formula and a Nevanlinna–Pick type theorem are obtained.

References

Jim Agler, On the representation of certain holomorphic functions defined on a polydisc, Topics in operator theory: Ernst D. Hellinger memorial volume, Oper. Theory Adv. Appl., vol. 48, Birkhäuser, Basel, 1990, pp. 47–66. MR 1207393
Jim Agler and John E. McCarthy, Nevanlinna-Pick interpolation on the bidisk, J. Reine Angew. Math. 506 (1999), 191–204. MR 1665697, DOI 10.1515/crll.1999.004
J. Agler and N. J. Young, A commutant lifting theorem for a domain in $\mathbf C^2$ and spectral interpolation, J. Funct. Anal. 161 (1999), no. 2, 452–477. MR 1674635, DOI 10.1006/jfan.1998.3362
G. D. Allen, F. J. Narcowich, and J. P. Williams, An operator version of a theorem of Kolmogorov, Pacific J. Math. 61 (1975), no. 2, 305–312. MR 405167
C.G. Ambrozie, M. Engliš and V. Müller: Operator tuples and analytic models over general domains in $\mathbb {C}^n$, J. Operator Theory, 42:2 (2002).
C.-G. Ambrozie, D. Timotin: On an intertwining lifting for certain reproducing kernel Hilbert spaces, Integral Equations Operator Theory, 42 (2002), 373–384.
T. Andô, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963), 88–90. MR 155193
William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
C. Badea, G. Cassier: Constrained von Neumann inequalities, Advances in Mathematics, to appear.
J. A. Ball, W. S. Li, D. Timotin, and T. T. Trent, A commutant lifting theorem on the polydisc, Indiana Univ. Math. J. 48 (1999), no. 2, 653–675. MR 1722812, DOI 10.1512/iumj.1999.48.1708
Joseph A. Ball and Tavan T. Trent, Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables, J. Funct. Anal. 157 (1998), no. 1, 1–61. MR 1637941, DOI 10.1006/jfan.1998.3278
J.A. Ball, T.T. Trent, V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, Operator Theory and Analysis: The M.A. Kaashoek Anniversary Volume (Workshop in Amsterdam, Nov. 1997), Operator Theory: Advances and Applications Vol. 122, Birkhäuser–Verlag, 2001, 89–138.
Robert S. Clancy and Scott McCullough, Projective modules and Hilbert spaces with a Nevanlinna-Pick kernel, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3299–3305. MR 1473659, DOI 10.1090/S0002-9939-98-04624-3
R. E. Curto and F.-H. Vasilescu, Standard operator models in the polydisc, Indiana Univ. Math. J. 42 (1993), no. 3, 791–810. MR 1254118, DOI 10.1512/iumj.1993.42.42035
Dmitriy S. Kalyuzhniy, Multiparametric dissipative linear stationary dynamical scattering systems: discrete case, J. Operator Theory 43 (2000), no. 2, 427–460. MR 1753418
V. Müller and F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), no. 4, 979–989. MR 1112498, DOI 10.1090/S0002-9939-1993-1112498-0
Z. Słodkowski and W. Żelazko, On joint spectra of commuting families of operators, Studia Math. 50 (1974), 127–148. MR 346555, DOI 10.4064/sm-50-2-127-148
Béla Sz.-Nagy and Ciprian Foiaş, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York; Akadémiai Kiadó, Budapest, 1970. Translated from the French and revised. MR 0275190
Andrew T. Tomerlin, Products of Nevanlinna-Pick kernels and operator colligations, Integral Equations Operator Theory 38 (2000), no. 3, 350–356. MR 1797710, DOI 10.1007/BF01291719
Harald Upmeier, Toeplitz operators and index theory in several complex variables, Operator Theory: Advances and Applications, vol. 81, Birkhäuser Verlag, Basel, 1996. MR 1384981, DOI 10.1016/s0168-9274(96)90019-7
Florian-Horia Vasilescu, Analytic functional calculus and spectral decompositions, Mathematics and its Applications (East European Series), vol. 1, D. Reidel Publishing Co., Dordrecht; Editura Academiei Republicii Socialiste România, Bucharest, 1982. Translated from the Romanian. MR 690957
Nicholas Th. Varopoulos, Sur une inégalité de von Neumann, C. R. Acad. Sci. Paris Sér. A-B 277 (1973), A19–A22 (French). MR 322555