Analytic models for commuting operator tuples on bounded symmetric domains

Authors:
Jonathan Arazy and Miroslav Englis

Journal:
Trans. Amer. Math. Soc. **355** (2003), 837-864

MSC (2000):
Primary 47A45; Secondary 47A13, 32M15, 32A07

Published electronically:
October 9, 2002

MathSciNet review:
1932728

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Abstract | References | Similar Articles | Additional Information

Abstract: For a domain in and a Hilbert space of analytic functions on which satisfies certain conditions, we characterize the commuting -tuples of operators on a separable Hilbert space such that is unitarily equivalent to the restriction of to an invariant subspace, where is the operator -tuple on the Hilbert space tensor product . For the unit disc and the Hardy space , this reduces to a well-known theorem of Sz.-Nagy and Foias; for a reproducing kernel Hilbert space on such that the reciprocal of its reproducing kernel is a polynomial in and , this is a recent result of Ambrozie, Müller and the second author. In this paper, we extend the latter result by treating spaces for which ceases to be a polynomial, or even has a pole: namely, the standard weighted Bergman spaces (or, rather, their analytic continuation) on a Cartan domain corresponding to the parameter in the continuous Wallach set, and reproducing kernel Hilbert spaces for which is a rational function. Further, we treat also the more general problem when the operator is replaced by , being a certain generalization of a unitary operator tuple. For the case of the spaces on Cartan domains, our results are based on an analysis of the homogeneous multiplication operators on , which seems to be of an independent interest.

**[Ag1]**Jim Agler,*The Arveson extension theorem and coanalytic models*, Integral Equations Operator Theory**5**(1982), no. 5, 608–631. MR**697007**, 10.1007/BF01694057**[Ag2]**Jim Agler,*Hypercontractions and subnormality*, J. Operator Theory**13**(1985), no. 2, 203–217. MR**775993****[AEM]**C. Ambrozie, M. Englis, V. Müller,*Operator tuples and analytic models over general domains in*, preprint (1999), J. Oper. Theory**47**(2002), 287-302.**[Ara]**Jonathan Arazy,*A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains*, Multivariable operator theory (Seattle, WA, 1993) Contemp. Math., vol. 185, Amer. Math. Soc., Providence, RI, 1995, pp. 7–65. MR**1332053**, 10.1090/conm/185/02147**[ArU]**J. Arazy, H. Upmeier,*Boundary integration and the discrete Wallach points*, ESI preprint No.**762**(1999),`http://www.esi.ac.at/preprints/ESI-Preprints.html`, to be submitted to AMS Memoirs.**[ArZ]**J. Arazy, G. Zhang,*Homogeneous multiplication operators on bounded symmetric domains*, preprint, 2001, submitted to J. Funct. Anal.**[Aro]**N. Aronszajn,*Theory of reproducing kernels*, Trans. Amer. Math. Soc.**68**(1950), 337–404. MR**0051437**, 10.1090/S0002-9947-1950-0051437-7**[Arv]**William Arveson,*Subalgebras of 𝐶*-algebras. III. Multivariable operator theory*, Acta Math.**181**(1998), no. 2, 159–228. MR**1668582**, 10.1007/BF02392585**[At1]**Ameer Athavale,*Holomorphic kernels and commuting operators*, Trans. Amer. Math. Soc.**304**(1987), no. 1, 101–110. MR**906808**, 10.1090/S0002-9947-1987-0906808-6**[At2]**Ameer Athavale,*On the intertwining of joint isometries*, J. Operator Theory**23**(1990), no. 2, 339–350. MR**1066811****[At3]**Ameer Athavale,*Model theory on the unit ball in 𝐶^{𝑚}*, J. Operator Theory**27**(1992), no. 2, 347–358. MR**1249651****[Be]**Stefan Bergman,*The kernel function and conformal mapping*, Second, revised edition, American Mathematical Society, Providence, R.I., 1970. Mathematical Surveys, No. V. MR**0507701****[Cu]**Raul E. Curto,*The spectra of elementary operators*, Indiana Univ. Math. J.**32**(1983), no. 2, 193–197. MR**690184**, 10.1512/iumj.1983.32.32017**[Dr]**S. W. Drury,*A generalization of von Neumann’s inequality to the complex ball*, Proc. Amer. Math. Soc.**68**(1978), no. 3, 300–304. MR**480362**, 10.1090/S0002-9939-1978-0480362-8**[FK]**J. Faraut and A. Korányi,*Function spaces and reproducing kernels on bounded symmetric domains*, J. Funct. Anal.**88**(1990), no. 1, 64–89. MR**1033914**, 10.1016/0022-1236(90)90119-6**[Fe]**Charles Fefferman,*The Bergman kernel and biholomorphic mappings of pseudoconvex domains*, Invent. Math.**26**(1974), 1–65. MR**0350069****[Gh]**C. Robin Graham,*Scalar boundary invariants and the Bergman kernel*, Complex analysis, II (College Park, Md., 1985–86) Lecture Notes in Math., vol. 1276, Springer, Berlin, 1987, pp. 108–135. MR**922320**, 10.1007/BFb0078958**[KS]**Friedrich Knop and Siddhartha Sahi,*A recursion and a combinatorial formula for Jack polynomials*, Invent. Math.**128**(1997), no. 1, 9–22. MR**1437493**, 10.1007/s002220050134**[La]**Michel Lassalle,*Une formule de Pieri pour les polynômes de Jack*, C. R. Acad. Sci. Paris Sér. I Math.**309**(1989), no. 18, 941–944 (French, with English summary). MR**1054739****[Lo]**O. Loos,*Bounded symmetric domains and Jordan pairs*, University of California, Irvine, 1977.**[MD]**I. G. Macdonald,*Symmetric functions and Hall polynomials*, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR**1354144****[M]**V. P. Maslov,*Operatornye metody*, Izdat. “Nauka”, Moscow, 1973 (Russian). MR**0445305****[MN]**V. P. Maslov and V. E. Nazaĭkinskiĭ,*Asymptotics of operator and pseudo-differential equations*, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1988. Translated from the Russian. MR**954389****[MV]**V. Müller and F.-H. Vasilescu,*Standard models for some commuting multioperators*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 979–989. MR**1112498**, 10.1090/S0002-9939-1993-1112498-0**[Po]**Sandra Pott,*Standard models under polynomial positivity conditions*, J. Operator Theory**41**(1999), no. 2, 365–389. MR**1681579****[RV]**M. Vergne and H. Rossi,*Analytic continuation of the holomorphic discrete series of a semi-simple Lie group*, Acta Math.**136**(1976), no. 1-2, 1–59. MR**0480883****[St]**Richard P. Stanley,*Some combinatorial properties of Jack symmetric functions*, Adv. Math.**77**(1989), no. 1, 76–115. MR**1014073**, 10.1016/0001-8708(89)90015-7**[Up]**Harald Upmeier,*Toeplitz operators on bounded symmetric domains*, Trans. Amer. Math. Soc.**280**(1983), no. 1, 221–237. MR**712257**, 10.1090/S0002-9947-1983-0712257-2**[Vr]**Lars Vretare,*Recurrence formulas for zonal polynomials*, Math. Z.**188**(1985), no. 3, 419–425. MR**771995**, 10.1007/BF01159186**[Zh]**Gen Kai Zhang,*Some recurrence formulas for spherical polynomials on tube domains*, Trans. Amer. Math. Soc.**347**(1995), no. 5, 1725–1734. MR**1249896**, 10.1090/S0002-9947-1995-1249896-7

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Additional Information

**Jonathan Arazy**

Affiliation:
Department of Mathematics, University of Haifa, Haifa 31905, Israel

Email:
jarazy@math.haifa.ac.il

**Miroslav Englis**

Affiliation:
MÚ AV ČR, Žitná 25, 11567 Prague 1, Czech Republic

Email:
englis@math.cas.cz

DOI:
http://dx.doi.org/10.1090/S0002-9947-02-03156-2

Keywords:
Coanalytic models,
reproducing kernels,
bounded symmetric domains,
commuting operator tuples,
functional calculus

Received by editor(s):
February 14, 2002

Published electronically:
October 9, 2002

Additional Notes:
The second author’s research was supported by GA ČR grant no. 201/00/0208 and GA AV ČR grant no. A1019005. The second author was also partially supported by the Israeli Academy of Sciences during his visit to Haifa in January 2001, during which part of this work was done.

Article copyright:
© Copyright 2002
American Mathematical Society