Abstract
In this chapter, we will apply those methods developed in Chaps. 3–6 to study similarity and unitary equivalence of multiplication operators, defined on both the Hardy space and the Bergman space.
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Bibliography
X. Chen, K. Guo, Analytic Hilbert Modules. π-Chapman & Hall/CRC Research Notes in Mathematics, vol. 433, (2003)
D. Clark, On Toeplitz operators with loops. J. Oper. Theory 4, 37–54 (1980)
D. Clark, On a similarity theory for rational Toeplitz operators. J. Riene Angew. Math. 320, 6–31 (1980)
D. Clark, On Toeplitz operators with loops II. J. Oper. Theory 7, 109–123 (1982)
D. Clark, Sz-Nagy-Foias theory and similarity for a class of Toeplitz operators, in Proceedings of the 1977 Spectral theory Semester (The Stefan Banach Mathematical Center, Warsaw, 1982)
D. Clark, J. Morrel, On Toeplitz operators and similarity. Am. J. Math. 100, 973–986 (1978)
M. Cowen, R. Douglas, Complex geometry and operator theory. Acta Math. 141, 187–261 (1978)
C. Cowen, The commutant of an analytic Toeplitz operator. Trans. Am. Math. Soc. 239, 1–31 (1978)
C. Cowen, The commutant of an analytic Toeplitz operator, II. Indiana Univ. Math. J. 29, 1–12 (1980)
C. Cowen, An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators. Funct. Anal. 36, 169–184 (1980)
C. Cowen, On equivalence of Toeplitz operators. J. Oper. Theory 7, 167–172 (1982)
C. Cowen, R. Wahl, Commutants of finite Blaschke product multiplication operators. Preprint
R. Douglas, Operator theory and complex geometry. Extracta Math. 24, 135–165 (2009). arXiv: math.FA/ 0710.1880v2
R. Douglas, V. Paulsen, Hilbert Modules over Function Algebras. Pitman Research Notes in Mathematics, vol. 217 (Longman Scientific & Technical, Harlow, 1989)
P. Duren, A. Schuster, Bergman Spaces. Mathematics Surveys and Monographs, vol. 100 (American Mathematical Society, Rhode Island, 2004)
P. Duren, Extension of a result of Beurling on invariant subspaces. Trans. Am. Math. Soc. 99, 320–324 (1961)
J. Garnett, Bounded Analytic Functions (Academic, New York, 1981)
K. Guo, H. Huang, On multiplication operators of the Bergman space: similarity, unitary equivalence and reducing subspaces. J. Oper. Theory 65, 355–378 (2011)
K. Guo, Algebraic reduction for Hardy submodules over polydisk algebras. J. Oper. Theory 41, 127–138 (1999)
K. Guo, Characteristic spaces and rigidity of analytic Hilbert modules. J. Funct. Anal. 163, 133–151 (1999)
K. Guo, Equivalence of Hardy submodules generated by polynomials. J. Funct. Anal. 178, 343–371 (2000)
K. Guo, Defect operators, defect functions and defect indices for analytic submodules. J. Funct. Anal. 213, 380–411 (2004)
K. Guo, Operator theory and Von Neumann algebras (a manuscript)
K. Guo, Defect operators for submodules of H d 2. J. Reine Angew. Math. 573, 181–209 (2004)
K. Guo, K. Wang, On operators which commute with analytic Toeplitz operators modulo the finite rank operators. Proc. Am. Math. Soc. 134, 2571–2576 (2006)
K. Guo, K. Wang, Essentially normal Hilbert modules and K-homology II: quasi-homogeneous Hilbert modules over the two dimensional unit ball. J. Ramanujan Math. Soc. 22, 259–281 (2007)
K. Guo, K. Wang, Essentially normal Hilbert modules and K-homology. Math. Ann. 340, 907–934 (2008)
K. Guo, K. Wang, Beurling type quotient modules over the bidisk and boundary representations. J. Funct. Anal. 257, 3218–3238 (2009)
K. Guo, D. Zheng, Rudin orthogonality problem on the Bergman space. J. Funct. Anal. 261, 51–68 (2011)
K. Hoffman, Banach Spaces of Analytic Functions (Prentice-Hall, Englewood Cliffs, 1962)
C. Horowitz, Zeros of functions in the Bergman spaces. Duke Math. J. 41, 693–710 (1974)
C. Jiang, Y. Li, The commutant and similarity invariant of analytic Toeplitz operators on Bergman space. Sci. China Ser. A 5, 651–664 (2007)
C. Jiang, D. Zheng, Similarity of analytic Toeplitz operators on the Bergman spaces. J. Funct. Anal. 258, 2961–2982 (2010)
V. Peller, Hankel Operators and Their Applications. Springer Monographs in Mathematics (Springer, New York, 2003)
W. Rudin, Function Theory in Polydiscs (Benjamin, New York, 1969)
A. Shields, Weighted Shift Operators and Analytic Function Theory. Mathematical Surveys, vol. 13 (American mathematical Society, Providence, 1974), 49–128
S. Sun, On unitary equivalence of multiplication operators on Bergman space. Northeast. Math. J. 1, 213–222 (1985)
S. Sun, D. Yu, On unitary equivalence of multiplication operators on Bergman space (II). Northeast. Math. J. 4, 169–179 (1988)
S. Sun, D. Zheng, C. Zhong, Classification of reducing subspaces of a class of multiplication operators via the Hardy space of the bidisk. Can. J. Math. 62, 415–438 (2010)
S. Sun, D. Zheng, C. Zhong, Multiplication operators on the Bergman space and weighted shifts. J. Oper. Theory 59, 435–452 (2008)
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Guo, K., Huang, H. (2015). Similarity and Unitary Equivalence. In: Multiplication Operators on the Bergman Space. Lecture Notes in Mathematics, vol 2145. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-46845-6_7
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