## Journal of the Mathematical Society of Japan

### The graded structure induced by operators on a Hilbert space

#### Abstract

In this paper we define a graded structure induced by operators on a Hilbert space. Then we introduce several concepts which are related to the graded structure and examine some of their basic properties. A theory concerning minimal property and unitary equivalence is then developed. It allows us to obtain a complete description of $\mathcal{V}^\ast(M_{z^k})$ on any $H^2(\omega)$. It also helps us to find that a multiplication operator induced by a quasi-homogeneous polynomial must have a minimal reducing subspace. After a brief review of multiplication operator $M_{z+w}$ on $H^2(\omega,\delta)$, we prove that the Toeplitz operator $T_{z+\overline{w}}$ on $H^2(\mathbb{D}^2)$, the Hardy space over the bidisk, is irreducible.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 2 (2018), 853-875.

Dates
First available in Project Euclid: 18 April 2018

https://projecteuclid.org/euclid.jmsj/1524038676

Digital Object Identifier
doi:10.2969/jmsj/07027503

Mathematical Reviews number (MathSciNet)
MR3787742

Zentralblatt MATH identifier
06902444

#### Citation

GUO, Kunyu; WANG, Xudi. The graded structure induced by operators on a Hilbert space. J. Math. Soc. Japan 70 (2018), no. 2, 853--875. doi:10.2969/jmsj/07027503. https://projecteuclid.org/euclid.jmsj/1524038676

#### References

• W. Arveson, The curvature of a Hilbert module over $\mathbb{C}[z_1,\ldots,z_d]$, Proc. Natl. Acad. Sci. USA, 96 (1999), 11096–11099.
• W. Arveson, The curvature invariant of a Hilbert module over $\mathbb{C}[z_1,\ldots,z_d]$, J. Reine Angew. Math., 522 (2000), 173–236.
• R. Bhatia and P. Rosenthal, How and why to solve the operator equation $AX-XB=Y$, Bull. London Math. Soc., 29 (1997), 1–21.
• L. Brown, R. Douglas and P. Fillmore, Unitary equivalence modulo the compact operators and extension of $C^\ast$-algebras, Proceedings of Conference on Operator Theory (Dalhousie Univ., Halifax, N. S.), Lecture Notes in Math., 345, Springer, Berlin, 1973, 58–128.
• J. Carlson, D. Clark, C. Foias and J. Williams, Projective Hilbert $A(\mathbb{D})$-modules, New York J. Math., 1 (1994), 26–38.
• J. Carlson and D. Clark, Cohomology and extensions of Hilbert modules, J. Funct. Anal., 128 (1995), 278–306.
• J. Carlson and D. Clark, Projectivity and extensions of Hilbert modules over $A(\mathbb{D}^N)$, Mich. Math. J., 44 (1997), 365–373.
• X. Chen and R. Douglas, Localization of Hilbert modules, Mich. Math. J., 39 (1992), 443–454.
• X. Chen and K. Guo, Analytic Hilbert Modules, Research Notes in Mathematics, 433, Chapman & Hal/CRC, 2003.
• J. Conway, A Course in Operator Theory, Graduate Studies in Mathematics, 21, AMS, Rhode Island, 2000.
• M. Cowen and R. Douglas, Complex geometry and operator theory, Acta Math., 141 (1978), 187–261.
• H. Dan and H. Huang, Multiplication operators defined by a class of polynomials on $L_a^2(\mathbb{D}^2)$, Integr. Equ. Oper. Theory, 80 (2014), 581–601.
• J. Deng, Y. Lu and Y. Shi, Reducing subspaces for a class of non-analytic Toeplitz operators on the bidisk, J. Math. Anal. Appl., 445 (2017), 784–796.
• R. Douglas and G. Misra, Equivalence of quotient Hilbert modules, Proc. Indian Acad. Sci. (Math. Sci.), 113 (2003), 281–291.
• R. Douglas and G. Misra, Equivalence of quotient Hilbert modules II, Trans. Am. Math. Soc., 360 (2008), 2229–2264.
• R. Douglas and V. Paulsen, Hilbert modules over function algebras, Pitman Research Notes in Mathematics Series, 217, Longman Scientific & Technical, Harlow, 1989.
• R. Douglas, V. Paulsen and K. Yan, Operator theory and algebraic geometry, Bull. Amer. Math. Soc., 20 (1989), 67–71.
• R. Douglas and K. Yan, Hilbert–Samuel polynomials for Hilbert modules, Indiana Univ. Math. J., 42 (1993), 811–820.
• Y. Duan and K. Guo, Dimension formula for localization of Hilbert modules, J. Oper. Theory, 62 (2009), 439–452.
• D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, 150, Springer, 2004.
• X. Fang, Samuel multiplicity and the structure of semi-Fredholm operators, Adv. Math., 186 (2004), 411–437.
• X. Fang, The Fredholm index of quotient Hilbert modules, Math. Res. Lett., 12 (2005), 911–920.
• K. Guo, Characteristic spaces and rigidity for analytic Hilbert modules, J. Funct. Anal., 163 (1999), 133–151.
• K. Guo and H. Huang, Multiplication Operators on the Bergman Space, Lecture Notes in Math., 2145, Springer, 2015.
• K. Guo and X. Wang, Reducing subspaces of tensor products of weighted shifts, Sci. China Math., 59 (2016), 715–730.
• Y. Lu and X. Zhou, Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk, J. Math. Soc. Japan, 62 (2010), 745–765.
• J. Sarkar, An introduction to Hilbert module approach to multivariable operator theory, Operator Theory, Springer, 2015, 969–1033.
• J. Sarkar, Applications of Hilbert module approach to multivariable operator theory, Operator Theory, Springer, 2015, 1035–1091.
• Y. Shi and Y. Lu, Reducing subspaces for Toeplitz operators on the polydisk, Bull. Korean Math. Soc., 50 (2013), 687–696.
• A. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Math. Surveys, 13 (1974), 49–128.
• M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc., 130 (2002), 2631–2639.
• X. Wang, H. Dan and H. Huang, Reducing subspaces of multiplication operators with the symbol $\alpha z^k+\beta w^l$ on $L_a^2(\mathbb{D}^2)$, Sci. China Math., 58 (2015), 2167–2180.