Abstract
One can view contraction operators given by a canonical model of Sz.-Nagy and Foias as being defined by a quotient module where the basic building blocks are Hardy spaces.I n this note we generalize this framework to allow the Bergman and weighted Bergman spaces as building blocks, but restricting attention to the case in which the operator obtained is in the Cowen-Douglas class and requiring the multiplicity to be one.W e view the classification of such operators in the context of complex geometry and obtain a complete classification up to unitary equivalence of them in terms of their associated vector bundles and their curvatures.
Mathematics Subject Classification (2000). 46E22, 46M20, 47A20, 47A45, 47B32.
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References
H. Bercovici, R.G. Douglas, and C. Foias, Canonical models for bi-isometries, Operator Theory: Advances and Applications, 218 (2011), 177-205.
H. Bercovici, R.G. Douglas, C. Foias, and C. Pearcy, Confluent operator algebras and the closability property, J. Funct. Anal. 258 (2010) 4122-4153.
X. Chen and R.G. Douglas, Localization of Hilbert modules, Mich. Math. J. 39 (1992), 443-454.
M.J. Cowen and R.G. Douglas, Complex geometry and operator theory, Acta Math. 141 (1978), 187-261.
R.G. Douglas, C. Foias, and J. Sarkar, Resolutions of Hilbert modules and similarity, Journal of Geometric Analysis, to appear.
R.G. Douglas, Y. Kim, H. Kwon, and J. Sarkar, Curvature invariant and generalized canonical operator models - II, in preparation.
R.G. Douglas and G. Misra, Quasi-free resolutions of Hilbert modules, Integral Equations Operator Theory 47 (2003), No. 4, 435-456.
R.G. Douglas, G. Misra, and J. Sarkar, Contractive Hilbert modules and their dilations over natural function algebras, Israel Journal of Math, to appear.
R.G. Douglas and V.I. Paulsen, Hilbert Modules over Function Algebras, Research Notes in Mathematics Series, 47, Longman, Harlow, 1989.
G. Polya, How to Solve It: A New Aspect of Mathematical Method, Princeton University Press, Princeton, 1944.
M.A. Shubin, Factorization of parameter-dependent matrix functions in normal rings and certain related questions in the theory of Noetherian operators, Mat. Sb. 73 (113) (1967) 610-629; Math. USSR Sb.
B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970.
M. Uchiyama, Curvatures and similarity of operators with holomorphic eigenvectors, Trans. Amer. Math. Soc. 319 (1990), 405-415.
K. Zhu, Operator Theory in Function Spaces, Mathematical Surveys and Monographs, 138, American Mathematical Society, Providence, 2007.
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Douglas, R.G., Kim, YS., Kwon, HK., Sarkar, J. (2012). Curvature Invariant and Generalized Canonical Operator Models – I. In: Arendt, W., Ball, J., Behrndt, J., Förster, KH., Mehrmann, V., Trunk, C. (eds) Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations. Operator Theory: Advances and Applications, vol 221. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-0297-0_16
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DOI: https://doi.org/10.1007/978-3-0348-0297-0_16
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