On the duals of Szegö and Cauchy tuples
Authors:
Ameer Athavale and Pramod Patil
Journal:
Proc. Amer. Math. Soc. 139 (2011), 491498
MSC (2010):
Primary 47B20; Secondary 33C55
Published electronically:
July 12, 2010
MathSciNet review:
2736332
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Abstract: The tuple of multiplications by coordinate functions on the Hardy space of the open unit ball in (resp. open unit polydisk in ) is referred to as the Szegö tuple (resp. Cauchy tuple) and is a wellknown example of a subnormal operator tuple. Naturally associated with the Szegö tuple (resp. Cauchy tuple) is its dual whose coordinates act on the orthocomplement of the Hardy space of the ball (resp. polydisk) in an appropriate space. We examine the Koszul complexes associated with the duals of the Szegö and Cauchy tuples and determine their Betti numbers. We explicitly verify that, for , the 'th cohomology vector space associated with the Koszul complex of either the dual of the Szegö tuple or the dual of the Cauchy tuple is zerodimensional. It follows in particular that, for , neither the Szegö tuple nor the Cauchy tuple is quasisimilar to its dual; this is in contrast with the case where both the Szegö tuple and the Cauchy tuple reduce to the Unilateral Shift, which is known to be unitarily equivalent to its dual.
 1.
Ameer
Athavale, On the duals of subnormal tuples, Integral Equations
Operator Theory 12 (1989), no. 3, 305–323. MR 998276
(90g:47042), http://dx.doi.org/10.1007/BF01235735
 2.
Ameer
Athavale, On the intertwining of joint isometries, J. Operator
Theory 23 (1990), no. 2, 339–350. MR 1066811
(91i:47029)
 3.
Ameer
Athavale, Quasisimilarityinvariance of joint spectra for certain
subnormal tuples, Bull. Lond. Math. Soc. 40 (2008),
no. 5, 759–769. MR 2439641
(2009j:47038), http://dx.doi.org/10.1112/blms/bdn054
 4.
CălinGrigore
Ambrozie and FlorianHoria
Vasilescu, Banach space complexes, Mathematics and its
Applications, vol. 334, Kluwer Academic Publishers Group, Dordrecht,
1995. MR
1357165 (97a:47001)
 5.
John
B. Conway, The dual of a subnormal operator, J. Operator
Theory 5 (1981), no. 2, 195–211. MR 617974
(84j:47037)
 6.
Raul
E. Curto, Fredholm and invertible
𝑛tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), no. 1, 129–159. MR 613789
(82g:47010), http://dx.doi.org/10.1090/S00029947198106137896
 7.
Raul
E. Curto, Spectral inclusion for doubly
commuting subnormal 𝑛tuples, Proc.
Amer. Math. Soc. 83 (1981), no. 4, 730–734. MR 630045
(82j:47030), http://dx.doi.org/10.1090/S00029939198106300456
 8.
Raúl
E. Curto, Applications of several complex variables to
multiparameter spectral theory, Surveys of some recent results in
operator theory, Vol.\ II, Pitman Res. Notes Math. Ser., vol. 192,
Longman Sci. Tech., Harlow, 1988, pp. 25–90. MR 976843
(90d:47007)
 9.
Raúl
E. Curto and Ke
Ren Yan, The spectral picture of Reinhardt measures, J. Funct.
Anal. 131 (1995), no. 2, 279–301. MR 1345033
(96i:47006), http://dx.doi.org/10.1006/jfan.1995.1090
 10.
Jörg
Eschmeier and Mihai
Putinar, Spectral decompositions and analytic sheaves, London
Mathematical Society Monographs. New Series, vol. 10, The Clarendon
Press, Oxford University Press, New York, 1996. Oxford Science
Publications. MR
1420618 (98h:47002)
 11.
Jim
Gleason, On a question of Ameer Athavale, Irish Math. Soc.
Bull. 48 (2002), 31–33. MR 1930523
(2003h:47041)
 12.
Jim
Gleason, Stefan
Richter, and Carl
Sundberg, On the index of invariant subspaces in spaces of analytic
functions of several complex variables, J. Reine Angew. Math.
587 (2005), 49–76. MR 2186975
(2006i:47013), http://dx.doi.org/10.1515/crll.2005.2005.587.49
 13.
Takasi
Itô, On the commutative family of subnormal operators,
J. Fac. Sci. Hokkaido Univ. Ser. I 14 (1958), 1–15.
MR
0107177 (21 #5902)
 14.
Walter
Rudin, Function theory in the unit ball of 𝐶ⁿ,
Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of
Mathematical Science], vol. 241, SpringerVerlag, New YorkBerlin,
1980. MR
601594 (82i:32002)
 1.
 A. Athavale, On the duals of subnormal tuples, Integral Equations and Operator Theory 12 (1989), 305323. MR 998276 (90g:47042)
 2.
 A. Athavale, On the intertwining of joint isometries, J. Operator Theory 23 (1990), 339350. MR 1066811 (91i:47029)
 3.
 A. Athavale, Quasisimilarityinvariance of joint spectra for certain subnormal tuples, Bull. London Math. Soc. 5 (2008), 759769. MR 2439641 (2009j:47038)
 4.
 C.G. Ambrozie and F.H. Vasilescu, Banach Space Complexes, Kluwer Academic Publications, Dordrecht, 1995. MR 1357165 (97a:47001)
 5.
 J. B. Conway, The dual of a subnormal operator, J. Operator Theory 5 (1981), 195211. MR 617974 (84j:47037)
 6.
 R. E. Curto, Fredholm and invertible tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), 129159. MR 613789 (82g:47010)
 7.
 R. E. Curto, Spectral inclusion for doubly commuting subnormal tuples, Proc. Amer. Math. Soc. 83 (1981), 730734. MR 630045 (82j:47030)
 8.
 R. E. Curto, Applications of several complex variables to multiparameter spectral theory, Surveys of some recent results in operator theory, Vol. II, eds. J. B. Conway and B. B. Morrel, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, UK (1988), 2590. MR 976843 (90d:47007)
 9.
 R. E. Curto and K. Yan, The spectral picture of Reinhardt measures, J. Func. Anal. 131 (1995), 279301. MR 1345033 (96i:47006)
 10.
 J. Eschmeier and M. Putinar, Spectral decompositions and analytic sheaves, London Mathematical Series Monographs, New Series, vol. 10, Clarendon Press, Oxford (1996). MR 1420618 (98h:47002)
 11.
 J. Gleason, On a question of Ameer Athavale, Irish Math. Soc. Bull. 48 (2002), 3133. MR 1930523 (2003h:47041)
 12.
 J. Gleason, S. Richter and C. Sundberg, On the index of invariant subspaces in spaces of analytic functions in several complex variables, J. Reine Angew. Math. 587 (2005), 4976. MR 2186975 (2006i:47013)
 13.
 T. Ito, On the commutative family of subnormal operators, J. Fac. Sci. Hokkaido Univ. 14 (1958), 115. MR 0107177 (21:5902)
 14.
 W. Rudin, Function theory in the unit ball of , SpringerVerlag, New York, 1980. MR 601594 (82i:32002)
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Additional Information
Ameer Athavale
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email:
athavale@math.iitb.ac.in
Pramod Patil
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India
Email:
pramodp@math.iitb.ac.in
DOI:
http://dx.doi.org/10.1090/S000299392010104823
PII:
S 00029939(2010)104823
Keywords:
Subnormal,
quasisimilar,
Koszul complex,
spherical harmonics
Received by editor(s):
July 15, 2009
Received by editor(s) in revised form:
March 3, 2010
Published electronically:
July 12, 2010
Communicated by:
Nigel J. Kalton
Article copyright:
© Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
