Analytic models for commuting operator tuples on bounded symmetric domains
Authors:
Jonathan Arazy and Miroslav Englis
Journal:
Trans. Amer. Math. Soc. 355 (2003), 837864
MSC (2000):
Primary 47A45; Secondary 47A13, 32M15, 32A07
Published electronically:
October 9, 2002
MathSciNet review:
1932728
Fulltext PDF Free Access
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 wellknown 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.
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C. Ambrozie, M. Englis, V. Müller, Operator tuples and analytic models over general domains in , preprint (1999), J. Oper. Theory 47 (2002), 287302.
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J. Arazy, A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains, Multivariable operator theory (R.E. Curto, R.G. Douglas, J.D. Pincus, N. Salinas, eds.), Contemporary Mathematics, vol. 185, Amer. Math. Soc., Providence, 1995, pp. 765. MR 96e:46034
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J. Arazy, H. Upmeier, Boundary integration and the discrete Wallach points, ESI preprint No. 762 (1999), http://www.esi.ac.at/preprints/ESIPreprints.html, to be submitted to AMS Memoirs.
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J. Arazy, G. Zhang, Homogeneous multiplication operators on bounded symmetric domains, preprint, 2001, submitted to J. Funct. Anal.
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N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337404. MR 14:479c
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W. Arveson, Subalgebras of algebras III: Multivariable operator theory, Acta Math. 181 (1998), 159228. MR 2000e:47013
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A. Athavale, Holomorphic kernels and commuting operators, Trans. Amer. Math. Soc. 304 (1987), 101110. MR 88m:47039
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A. Athavale, On the intertwining of joint isometries, J. Oper. Theory 23 (1990), 339350. MR 91i:47029
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A. Athavale, Model theory on the unit ball in , J. Oper. Theory 27 (1992), 347358. MR 94i:47011
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S. Bergman, The kernel function and conformal mapping, 2nd edition, AMS, Providence, 1970. MR 58:22502
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R. E. Curto, The spectra of elementary operators, Indiana Univ. Math. J. 32 (1983), 193197. MR 84e:47005
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S. W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), 300304. MR 80c:47010
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J. Faraut, A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 6489. MR 90m:32049
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C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 165. MR 50:2562
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C. R. Graham, Scalar boundary invariants and the Bergman kernel, Complex Analysis, II (College Park, MD, 198586), Lecture Notes in Math., vol. 1276, Springer, Berlin, 1987, pp. 108135. MR 89d:32045
 [KS]
F. Knop, S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 922. MR 98k:33040
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M. Lassalle, Une formule de Pieri pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 941944 (French). MR 91c:05192
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O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
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I. G. MacDonald, Symmetric functions and Hall polynomials, 2nd edition, Clarendon Press, Oxford, 1995. MR 96h:05207
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V. P. Maslov, Operator Methods, Nauka, Moscow, 1973 (in Russian); English translation: Operational Methods, Mir, Moscow, 1976. MR 56:3647
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V. P. Maslov, V. E. Nazaikinskii, Asymptotics of Operators and PseudoDifferential Equations, Consultants Bureau, New YorkLondon, 1988. MR 89i:58141
 [MV]
V. Müller, F.H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), 979989. MR 93e:47016
 [Po]
S. Pott, Standard models under polynomial positivity conditions, J. Oper. Theory 41 (1999), 365389. MR 2000j:47019
 [RV]
H. Rossi, M. Vergne, Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math. 136 (1976), 159. MR 58:1032
 [St]
R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76115. MR 90g:05020
 [Up]
H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221237. MR 85g:47042
 [Vr]
L. Vretare, Recurrence formulas for zonal polynomials, Math. Z. 188 (1985), 419425. MR 86j:22012
 [Zh]
G. Zhang, Some recurrence formulas for spherical polynomials on tube domains, Trans. Amer. Math. Soc. 347 (1995), 17251734. MR 95h:22018
 [Ag1]
 J. Agler, The Arveson extension theorem and coanalytic models, Integ. Equations Oper. Theory 5 (1982), 608631. MR 84g:47011
 [Ag2]
 J. Agler, Hypercontractions and subnormality, J. Oper. Theory 13 (1985), 203217. MR 86i:47028
 [AEM]
 C. Ambrozie, M. Englis, V. Müller, Operator tuples and analytic models over general domains in , preprint (1999), J. Oper. Theory 47 (2002), 287302.
 [Ara]
 J. Arazy, A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains, Multivariable operator theory (R.E. Curto, R.G. Douglas, J.D. Pincus, N. Salinas, eds.), Contemporary Mathematics, vol. 185, Amer. Math. Soc., Providence, 1995, pp. 765. MR 96e:46034
 [ArU]
 J. Arazy, H. Upmeier, Boundary integration and the discrete Wallach points, ESI preprint No. 762 (1999), http://www.esi.ac.at/preprints/ESIPreprints.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), 337404. MR 14:479c
 [Arv]
 W. Arveson, Subalgebras of algebras III: Multivariable operator theory, Acta Math. 181 (1998), 159228. MR 2000e:47013
 [At1]
 A. Athavale, Holomorphic kernels and commuting operators, Trans. Amer. Math. Soc. 304 (1987), 101110. MR 88m:47039
 [At2]
 A. Athavale, On the intertwining of joint isometries, J. Oper. Theory 23 (1990), 339350. MR 91i:47029
 [At3]
 A. Athavale, Model theory on the unit ball in , J. Oper. Theory 27 (1992), 347358. MR 94i:47011
 [Be]
 S. Bergman, The kernel function and conformal mapping, 2nd edition, AMS, Providence, 1970. MR 58:22502
 [Cu]
 R. E. Curto, The spectra of elementary operators, Indiana Univ. Math. J. 32 (1983), 193197. MR 84e:47005
 [Dr]
 S. W. Drury, A generalization of von Neumann's inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), 300304. MR 80c:47010
 [FK]
 J. Faraut, A. Koranyi, Function spaces and reproducing kernels on bounded symmetric domains, J. Funct. Anal. 88 (1990), 6489. MR 90m:32049
 [Fe]
 C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 165. MR 50:2562
 [Gh]
 C. R. Graham, Scalar boundary invariants and the Bergman kernel, Complex Analysis, II (College Park, MD, 198586), Lecture Notes in Math., vol. 1276, Springer, Berlin, 1987, pp. 108135. MR 89d:32045
 [KS]
 F. Knop, S. Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), 922. MR 98k:33040
 [La]
 M. Lassalle, Une formule de Pieri pour les polynômes de Jack, C. R. Acad. Sci. Paris Sér. I Math. 309 (1989), 941944 (French). MR 91c:05192
 [Lo]
 O. Loos, Bounded symmetric domains and Jordan pairs, University of California, Irvine, 1977.
 [MD]
 I. G. MacDonald, Symmetric functions and Hall polynomials, 2nd edition, Clarendon Press, Oxford, 1995. MR 96h:05207
 [M]
 V. P. Maslov, Operator Methods, Nauka, Moscow, 1973 (in Russian); English translation: Operational Methods, Mir, Moscow, 1976. MR 56:3647
 [MN]
 V. P. Maslov, V. E. Nazaikinskii, Asymptotics of Operators and PseudoDifferential Equations, Consultants Bureau, New YorkLondon, 1988. MR 89i:58141
 [MV]
 V. Müller, F.H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), 979989. MR 93e:47016
 [Po]
 S. Pott, Standard models under polynomial positivity conditions, J. Oper. Theory 41 (1999), 365389. MR 2000j:47019
 [RV]
 H. Rossi, M. Vergne, Analytic continuation of the holomorphic discrete series of a semisimple Lie group, Acta Math. 136 (1976), 159. MR 58:1032
 [St]
 R. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), 76115. MR 90g:05020
 [Up]
 H. Upmeier, Toeplitz operators on bounded symmetric domains, Trans. Amer. Math. Soc. 280 (1983), 221237. MR 85g:47042
 [Vr]
 L. Vretare, Recurrence formulas for zonal polynomials, Math. Z. 188 (1985), 419425. MR 86j:22012
 [Zh]
 G. Zhang, Some recurrence formulas for spherical polynomials on tube domains, Trans. Amer. Math. Soc. 347 (1995), 17251734. MR 95h:22018
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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/S0002994702031562
PII:
S 00029947(02)031562
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
