The General Theory of ${\cal{S}}_{T}$-Graded Modules and Kernel Method

Abstract

In this paper, the authors develop the theory of ${\cal{S}}_{T}$-graded modules to study some kinds of graded structures on reducing subspaces. They introduce some concepts and find an effective way to study ${\cal{S}}_{T}$-graded modules, which they name the kernel method. For its application, they define standard models for some multiplication operators and special Toeplitz operators, and show that these standard models can be solved by the kernel method. The authors also generalize the kernel method to arbitrary Hilbert spaces without any ${\cal{S}}_{T}$-graded condition. Finally, for any bounded operator T, they can find a reducing subspace that is ${\cal{S}}_{T}$-graded.

Keywords

Reducing subspace / Graded structure / Multiplication operator / Toeplitz operator / 47B37 / 47C15 / 47A80

References

[1] Albaseer M, Lu Y, Shi Y. Reducing subspaces for a class of Toeplitz operators on the Bergman space of the bidisk. Bull. Korean Math. Soc., 2015, 52(5): 1649-1660

[2] Dan H, Huang H. Multiplication operators defined by a class of polynomials on L_a²(ⅅ²). Integr. Equ. Oper. Theory, 2014, 80: 581-601

[3] Deng J, Lu Y, Shi Y. Reducing subspaces for a class of non-analytic Toeplitz operators on the bidisk. J. Math. Anal. Appl., 2017, 445(1): 784-796

[4] Douglas R, Putinar M, Wang K. Reducing subspaces for analytic multipliers of the Bergman space. J. Funct. Anal., 2012, 263: 1744-1765

[5] Douglas R, Sun S, Zheng D. Multiplication operators on the Bergman space via analytic continuation. Adv. Math., 2011, 226: 541-583

[6] Gu C. Reducing subspaces of non-analytic Toeplitz operators on weighted Hardy and Dirichlet spaces of the bidisk. J. Math. Anal. Appl., 2018, 459(2): 980-996

[7] Guo K, Huang H. On multiplication operators of the Bergman space: Similarity, unitary equivalence and reducing subspaces. J. Operator Theory, 2011, 65: 355-378

[8] Guo K, Huang H. Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras. J. Funct. Anal., 2011, 260: 1219-1255

[9] Guo K, Huang H. Reducing subspaces of multiplication operators on function spaces: Dedicated to the memory of Chen Kien-Kwong on the 120th anniversary of his birth. Appl. Math. J. Chinese Univ., 2013, 28: 395-404

[10] Guo K, Huang H. Geometric constructions of thin Blaschke products and reducing subspace problem. Proc. Lond. Math. Soc., 2014, 109: 1050-1091

[11] Guo K, Huang H. Multiplication Operators on the Bergman Space, 2015, Berlin, Heidelberg, Springer-Verlag2145

[12] Guo K, Sun S, Zheng D, Zhong C. Multiplication operators on the Bergman space via the Hardy space of the bidisk. J. Reine Angew. Math., 2009, 629: 129-168

[13] Guo K, Wang X. Reducing subspaces of tensor products of weighted shift. Sci. China Math., 2016, 59: 715-730

[14] Guo K, Wang X. The graded structure induced by operators on a Hilbert space. J. Math. Soc. Japan, 2018, 70(2): 853-875

[15] Hu J, Sun S, Xu X, Yu D. Reducing subspace of analytic Toeplitz operators on the Bergman space. Integr. Equ. Oper. Theory, 2004, 49: 387-395

[16] Lu Y, Zhou X. Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk. J. Math. Soc. Japan, 2010, 62: 745-765

[17] Shi Y, Lu Y. Reducing subspaces for Toeplitz operators on the polydisk. Bull. Korean Math. Soc., 2013, 50: 687-696

[18] Shields A. Weighted shift operators and analytic function theory, Topics in operator theory. Math. Surveys, 1974, 13: 49-128

[19] Stessin M, Zhu K. Reducing subspaces of weighted shift operators. Proc. Amer. Math. Soc., 2002, 130: 2631-2639

[20] Sun S, Zheng D, Zhong C. Multiplication operators on the Bergman space and weighted shifts. J. Operator Theory, 2008, 59: 435-452

[21] Sun S, Zheng D, Zhong C. Classification of reducing subspaces of a class of multiplication operators via the Hardy space of the bidisk. Canad. J. Math., 2010, 62: 415-438

[22] Wang X, Dan H, Huang H. Reducing subspaces of multiplication operators with the symbol αz^k + βw^l on L_a²(ⅅ²). Sci. China Math., 2015, 58: 2167-2180

[23] Zhu K. Reducing subspaces for a class of multiplication operators. J. London Math. Soc., 2000, 62: 553-568

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