# REDUCING SUBSPACES OF WEIGHTED SHIFTS WITH OPERATOR WEIGHTS

• Gu, Caixing (Department of Mathematics California Polytechnic State University)
• Published : 2016.09.30

#### Abstract

We characterize reducing subspaces of weighted shifts with operator weights as wandering invariant subspaces of the shifts with additional structures. We show how some earlier results on reducing subspaces of powers of weighted shifts with scalar weights on the unit disk and the polydisk can be fitted into our general framework.

#### References

1. M. B. Abrahamse and J. A. Ball, Analytic Toeplitz operators with automorphic symbol II, Proc. Amer. Math. Soc. 59 (1976), no. 2, 323-328. https://doi.org/10.1090/S0002-9939-1976-0454714-4
2. K. Guo and H. Huang, Multiplication operators on the Bergman space, Lecture Notes in Mathematics 2145, Springer, 2015.
3. P. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102-112.
4. Z. J. Jab lonski, I. B. Jung, and J. Stochel, Weighted shifts on directed trees, Mem. Amer. Math. Soc. 216 (2012), no. 1017, viii+106 pp.
5. N. P. Jewell and A. R. Lubin, Commuting weighted shifts and analytic function theory in several variables, J. Operator Theory 1 (1979), no. 2, 207-223.
6. Y. Lu and X. Zhou, Invariant subspaces and reducing subspaces of weighted Bergman space over bidisk, J. Math. Soc. Japan 62 (2010), no. 3, 745-765. https://doi.org/10.2969/jmsj/06230745
7. E. Nordgren, Reducing subspaces of analytic Toeplitz operators, Duke Math. J. 34 (1967), 175-181. https://doi.org/10.1215/S0012-7094-67-03419-9
8. H. Radjavi and P. Rosenthal, Simultaneous Triangularization, Universitext, Springer-Verlag, 2000.
9. Y. Shi and Y. Lu, Reducing subspaces for Toeplitz operators on the polydisk, Bull. Korean Math. Soc. 50 (2013), no. 2, 687-696. https://doi.org/10.4134/BKMS.2013.50.2.687
10. A. L. Shields, Weighted shift operators and analytic function theory, pp. 49-128 in Math. Surv. 13, Amer. Math. Soc., Providence, 1974.
11. M. Stessin and K. Zhu, Reducing subspaces of weighted shift operators, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2631-2639. https://doi.org/10.1090/S0002-9939-02-06382-7
12. K. Zhu, Reducing subspaces for a class of multiplication operators, J. London Math. Soc. 62 (2000), no. 2, 553-568. https://doi.org/10.1112/S0024610700001198