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Reducing subspaces on the annulus. (English) Zbl 1257.47011
Summary: We study reducing subspaces for an analytic multiplication operator $${M_{z^{n}}}$$ on the Bergman space $${L_{a}^{2}(A_{r})}$$ of the annulus $$A_{r}$$, and we prove that $${M_{z^{n}}}$$ has exactly $$2^{n }$$ reducing subspaces. Furthermore, in contrast to what happens for the disk, the same is true for the Hardy space on the annulus. Finally, we extend the results to certain bilateral weighted shifts, and interpret the results in the context of complex geometry.

##### MSC:
 47A15 Invariant subspaces of linear operators 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47B38 Linear operators on function spaces (general) 51D25 Lattices of subspaces and geometric closure systems
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##### References:
 [1] Ball J.A.: Hardy space expectation operators and reducing subspaces. Proc. Am. Math. Soc. 47, 351–357 (1975) · Zbl 0296.47022 · doi:10.1090/S0002-9939-1975-0358421-7 [2] Cowen C.C.: Iteration and the solution of functional equations for functions analytic in the unit disc. TAMS 265, 69–95 (1971) · Zbl 0476.30017 · doi:10.1090/S0002-9947-1981-0607108-9 [3] Cowen M.J., Douglas R.G.: Complex geometry and operator theory. Acta Math. 141, 188–261 (1978) · Zbl 0427.47016 · doi:10.1007/BF02545748 [4] Davidson K.R., Douglas R.G.: The generalized Berezin transform and commutator ideal. Pac. J. Math. 222(1), 29–56 (2005) · Zbl 1106.46042 · doi:10.2140/pjm.2005.222.29 [5] Douglas R.G.: Banach algebra techniques in operator theory. Springer, New York (1998) · Zbl 0920.47001 [6] Duren P., Schuster A.: Bergman Spaces. American Mathematical Society, Providence, RI (2004) [7] Jewell N.P.: Multiplication by the coordinate functions on the hardy space of the unit sphere in $${{$$\backslash$$mathbb{C}}\^n}$$ . Duke Math. J. 44, 839–851 (1977) · Zbl 0372.47016 · doi:10.1215/S0012-7094-77-04437-4 [8] Nordgren E.: Reducing subspaces of analytic Toeplitz operators. Duke Math. J. 34, 175–181 (1967) · Zbl 0184.35202 · doi:10.1215/S0012-7094-67-03419-9 [9] Shields, A.L.: Weighted Shift Operators and Analytic Function Theory. Mathematical Survey Series 13 (1974) · Zbl 0303.47021 [10] Stessin M., Zhu K.: Reducing subspaces of weighted shift operators. Proc. Am. Math. Soc. 130, 2631–2639 (2002) · Zbl 1035.47015 · doi:10.1090/S0002-9939-02-06382-7 [11] Thomson J.: The commutant of certain analytic Toeplitz operators. Proc. Am. Math. Soc. 54, 165–169 (1976) · Zbl 0328.47014 · doi:10.1090/S0002-9939-1976-0388156-7 [12] Zhu K.: Reducing subspaces for a class of multiplication operators. J. Lond. Math. Soc. 62, 553–568 (2000) · Zbl 1158.47309 · doi:10.1112/S0024610700001198
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