Berezin transforms on noncommutative polydomains
HTML articles powered by AMS MathViewer
- by Gelu Popescu PDF
- Trans. Amer. Math. Soc. 368 (2016), 4357-4416 Request permission
Abstract:
This paper is an attempt to unify the multivariable operator model theory for ball-like domains and commutative polydiscs and extend it to a more general class of noncommutative polydomains $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ in $B(\mathcal {H})^n$. An important role in our study is played by noncommutative Berezin transforms associated with the elements of the polydomain. These transforms are used to prove that each such polydomain has a universal model $\mathbf {W}=\{\mathbf {W}_{i,j}\}$ consisting of weighted shifts acting on a tensor product of full Fock spaces. We introduce the noncommutative Hardy algebra $F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})$ as the weakly closed algebra generated by $\{\mathbf {W}_{i,j}\}$ and the identity, and use it to provide a WOT-continuous functional calculus for completely non-coisometric tuples in $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$. It is shown that the Berezin transform is a completely isometric isomorphism between $F^\infty (\mathbf {D}_{\mathbf {q}}^{\mathbf {m}})$ and the algebra of bounded free holomorphic functions on the radial part of $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$. A characterization of the Beurling type joint invariant subspaces under $\{\mathbf {W}_{i,j}\}$ is also provided.
It has been an open problem for quite some time to find significant classes of elements in the commutative polydisc for which a theory of characteristic functions and model theory can be developed along the lines of the Sz.-Nagy–Foias theory of contractions. We give a positive answer to this question, in our more general setting, providing a characterization for the class of tuples of operators in $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$ which admit characteristic functions. The characteristic function is constructed explicitly as an artifact of the noncommutative Berezin kernel associated with the polydomain, and it is proved to be a complete unitary invariant for the class of completely non-coisometric tuples. Using noncommutative Berezin transforms and $C^*$-algebra techniques, we develop a dilation theory on the noncommutative polydomain $\mathbf {D}_{\mathbf {q}}^{\mathbf {m}}(\mathcal {H})$.
References
- Jim Agler, The Arveson extension theorem and coanalytic models, Integral Equations Operator Theory 5 (1982), no. 5, 608–631. MR 697007, DOI 10.1007/BF01694057
- Jim Agler, Hypercontractions and subnormality, J. Operator Theory 13 (1985), no. 2, 203–217. MR 775993
- William B. Arveson, Subalgebras of $C^{\ast }$-algebras, Acta Math. 123 (1969), 141–224. MR 253059, DOI 10.1007/BF02392388
- William Arveson, An invitation to $C^*$-algebras, Graduate Texts in Mathematics, No. 39, Springer-Verlag, New York-Heidelberg, 1976. MR 0512360, DOI 10.1007/978-1-4612-6371-5
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- William Arveson, The curvature invariant of a Hilbert module over $\textbf {C}[z_1,\cdots ,z_d]$, J. Reine Angew. Math. 522 (2000), 173–236. MR 1758582, DOI 10.1515/crll.2000.037
- Ameer Athavale, Holomorphic kernels and commuting operators, Trans. Amer. Math. Soc. 304 (1987), no. 1, 101–110. MR 906808, DOI 10.1090/S0002-9947-1987-0906808-6
- Joseph A. Ball and Victor Vinnikov, Lax-Phillips scattering and conservative linear systems: a Cuntz-algebra multidimensional setting, Mem. Amer. Math. Soc. 178 (2005), no. 837, iv+101. MR 2172325, DOI 10.1090/memo/0837
- T. Bhattacharyya, J. Eschmeier, and J. Sarkar, Characteristic function of a pure commuting contractive tuple, Integral Equations Operator Theory 53 (2005), no. 1, 23–32. MR 2183594, DOI 10.1007/s00020-004-1309-5
- T. Bhattacharyya and J. Sarkar, Characteristic function for polynomially contractive commuting tuples, J. Math. Anal. Appl. 321 (2006), no. 1, 242–259. MR 2236555, DOI 10.1016/j.jmaa.2005.07.075
- Chafiq Benhida and Dan Timotin, Automorphism invariance properties for certain families of multioperators, Operator theory live, Theta Ser. Adv. Math., vol. 12, Theta, Bucharest, 2010, pp. 5–15. MR 2731860
- Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239–255. MR 27954, DOI 10.1007/BF02395019
- F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 1134–1167 (Russian). MR 0350504
- John W. Bunce, Models for $n$-tuples of noncommuting operators, J. Funct. Anal. 57 (1984), no. 1, 21–30. MR 744917, DOI 10.1016/0022-1236(84)90098-3
- S. Brehmer, Über vetauschbare Kontraktionen des Hilbertschen Raumes, Acta Sci. Math. (Szeged) 22 (1961), 106–111 (German). MR 131169
- R. E. Curto and F.-H. Vasilescu, Automorphism invariance of the operator-valued Poisson transform, Acta Sci. Math. (Szeged) 57 (1993), no. 1-4, 65–78. MR 1243269
- 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
- R. E. Curto and F. H. Vasilescu, Standard operator models in the polydisc. II, Indiana Univ. Math. J. 44 (1995), no. 3, 727–746. MR 1375346, DOI 10.1512/iumj.1995.44.2005
- Kenneth R. Davidson and David R. Pitts, The algebraic structure of non-commutative analytic Toeplitz algebras, Math. Ann. 311 (1998), no. 2, 275–303. MR 1625750, DOI 10.1007/s002080050188
- Kenneth R. Davidson, David W. Kribs, and Miron E. Shpigel, Isometric dilations of non-commuting finite rank $n$-tuples, Canad. J. Math. 53 (2001), no. 3, 506–545. MR 1827819, DOI 10.4153/CJM-2001-022-0
- 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, DOI 10.1090/S0002-9939-1978-0480362-8
- Arthur E. Frazho, Models for noncommuting operators, J. Functional Analysis 48 (1982), no. 1, 1–11. MR 671311, DOI 10.1016/0022-1236(82)90057-X
- Steven G. Krantz, Function theory of several complex variables, AMS Chelsea Publishing, Providence, RI, 2001. Reprint of the 1992 edition. MR 1846625, DOI 10.1090/chel/340
- Vladimír Müller, Models for operators using weighted shifts, J. Operator Theory 20 (1988), no. 1, 3–20. MR 972177
- 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
- Paul S. Muhly and Baruch Solel, Tensor algebras over $C^*$-correspondences: representations, dilations, and $C^*$-envelopes, J. Funct. Anal. 158 (1998), no. 2, 389–457. MR 1648483, DOI 10.1006/jfan.1998.3294
- Paul S. Muhly and Baruch Solel, Hardy algebras, $W^\ast$-correspondences and interpolation theory, Math. Ann. 330 (2004), no. 2, 353–415. MR 2089431, DOI 10.1007/s00208-004-0554-x
- Paul S. Muhly and Baruch Solel, Canonical models for representations of Hardy algebras, Integral Equations Operator Theory 53 (2005), no. 3, 411–452. MR 2186099, DOI 10.1007/s00020-005-1373-5
- Anders Olofsson, A characteristic operator function for the class of $n$-hypercontractions, J. Funct. Anal. 236 (2006), no. 2, 517–545. MR 2240173, DOI 10.1016/j.jfa.2006.03.004
- Anders Olofsson, An operator-valued Berezin transform and the class of $n$-hypercontractions, Integral Equations Operator Theory 58 (2007), no. 4, 503–549. MR 2329133, DOI 10.1007/s00020-007-1502-4
- Vern I. Paulsen, Completely bounded maps and dilations, Pitman Research Notes in Mathematics Series, vol. 146, Longman Scientific & Technical, Harlow; John Wiley & Sons, Inc., New York, 1986. MR 868472
- Gilles Pisier, Similarity problems and completely bounded maps, Second, expanded edition, Lecture Notes in Mathematics, vol. 1618, Springer-Verlag, Berlin, 2001. Includes the solution to “The Halmos problem”. MR 1818047, DOI 10.1007/b55674
- Gelu Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), no. 2, 523–536. MR 972704, DOI 10.1090/S0002-9947-1989-0972704-3
- Gelu Popescu, Characteristic functions for infinite sequences of noncommuting operators, J. Operator Theory 22 (1989), no. 1, 51–71. MR 1026074
- Gelu Popescu, von Neumann inequality for $(B({\scr H})^n)_1$, Math. Scand. 68 (1991), no. 2, 292–304. MR 1129595, DOI 10.7146/math.scand.a-12363
- Gelu Popescu, Functional calculus for noncommuting operators, Michigan Math. J. 42 (1995), no. 2, 345–356. MR 1342494, DOI 10.1307/mmj/1029005232
- Gelu Popescu, Multi-analytic operators on Fock spaces, Math. Ann. 303 (1995), no. 1, 31–46. MR 1348353, DOI 10.1007/BF01460977
- Gelu Popescu, Poisson transforms on some $C^*$-algebras generated by isometries, J. Funct. Anal. 161 (1999), no. 1, 27–61. MR 1670202, DOI 10.1006/jfan.1998.3346
- Gelu Popescu, Curvature invariant for Hilbert modules over free semigroup algebras, Adv. Math. 158 (2001), no. 2, 264–309. MR 1822685, DOI 10.1006/aima.2000.1972
- Gelu Popescu, Free holomorphic functions on the unit ball of $B(\scr H)^n$, J. Funct. Anal. 241 (2006), no. 1, 268–333. MR 2264252, DOI 10.1016/j.jfa.2006.07.004
- Gelu Popescu, Operator theory on noncommutative varieties, Indiana Univ. Math. J. 55 (2006), no. 2, 389–442. MR 2225440, DOI 10.1512/iumj.2006.55.2771
- Gelu Popescu, Noncommutative Berezin transforms and multivariable operator model theory, J. Funct. Anal. 254 (2008), no. 4, 1003–1057. MR 2381202, DOI 10.1016/j.jfa.2007.06.004
- Gelu Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc. 200 (2009), no. 941, vi+91. MR 2519137, DOI 10.1090/memo/0941
- Gelu Popescu, Noncommutative transforms and free pluriharmonic functions, Adv. Math. 220 (2009), no. 3, 831–893. MR 2483229, DOI 10.1016/j.aim.2008.09.019
- Gelu Popescu, Free holomorphic functions on the unit ball of $B(\scr H)^n$. II, J. Funct. Anal. 258 (2010), no. 5, 1513–1578. MR 2566311, DOI 10.1016/j.jfa.2009.10.014
- Gelu Popescu, Operator theory on noncommutative domains, Mem. Amer. Math. Soc. 205 (2010), no. 964, vi+124. MR 2643314, DOI 10.1090/S0065-9266-09-00587-0
- Gelu Popescu, Free holomorphic automorphisms of the unit ball of $B(\scr H)^n$, J. Reine Angew. Math. 638 (2010), 119–168. MR 2595338, DOI 10.1515/CRELLE.2010.005
- Gelu Popescu, Joint similarity to operators in noncommutative varieties, Proc. Lond. Math. Soc. (3) 103 (2011), no. 2, 331–370. MR 2821245, DOI 10.1112/plms/pdr005
- Gelu Popescu, Free biholomorphic classification of noncommutative domains, Int. Math. Res. Not. IMRN 4 (2011), 784–850. MR 2773331, DOI 10.1093/imrn/rnq093
- Gelu Popescu, Berezin transforms on noncommutative varieties in polydomains, J. Funct. Anal. 265 (2013), no. 10, 2500–2552. MR 3091823, DOI 10.1016/j.jfa.2013.07.015
- Sandra Pott, Standard models under polynomial positivity conditions, J. Operator Theory 41 (1999), no. 2, 365–389. MR 1681579
- Walter Rudin, Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0255841
- W. Forrest Stinespring, Positive functions on $C^*$-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, DOI 10.1090/S0002-9939-1955-0069403-4
- Béla Sz.-Nagy, Ciprian Foias, Hari Bercovici, and László Kérchy, Harmonic analysis of operators on Hilbert space, Revised and enlarged edition, Universitext, Springer, New York, 2010. MR 2760647, DOI 10.1007/978-1-4419-6094-8
- Dan Timotin, Regular dilations and models for multicontractions, Indiana Univ. Math. J. 47 (1998), no. 2, 671–684. MR 1647873, DOI 10.1512/iumj.1998.47.1372
- Florian-Horia Vasilescu, An operator-valued Poisson kernel, J. Funct. Anal. 110 (1992), no. 1, 47–72. MR 1190419, DOI 10.1016/0022-1236(92)90042-H
- Johann von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951), 258–281 (German). MR 43386, DOI 10.1002/mana.3210040124
Additional Information
- Gelu Popescu
- Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
- MR Author ID: 234950
- Email: gelu.popescu@utsa.edu
- Received by editor(s): November 6, 2013
- Received by editor(s) in revised form: April 29, 2014
- Published electronically: September 15, 2015
- Additional Notes: This research was supported in part by an NSF grant
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4357-4416
- MSC (2010): Primary 46L52, 47A56; Secondary 47A48, 47A60
- DOI: https://doi.org/10.1090/tran/6466
- MathSciNet review: 3453374