Shift-invariant subspaces invariant for composition operators on the Hardy-Hilbert space

Cowen, Carl; Wahl, Rebecca

doi:10.1090/S0002-9939-2014-12132-0

Shift-invariant subspaces invariant for composition operators on the Hardy-Hilbert space
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by Carl C. Cowen and Rebecca G. Wahl PDF

Proc. Amer. Math. Soc. 142 (2014), 4143-4154 Request permission

Abstract:

If $\varphi$ is an analytic map of the unit disk $\mathbb {D}$ into itself, the composition operator $C_{\varphi }$ on a Hardy space $H^{2}$ is defined by $C_{\varphi }(f)=f\circ \varphi$. The unilateral shift on $H^{2}$ is the operator of multiplication by $z$. Beurling (1949) characterized the invariant subspaces for the shift. In this paper, we consider the shift-invariant subspaces that are invariant for composition operators. More specifically, necessary and sufficient conditions are provided for an atomic inner function with a single atom to be invariant for a composition operator, and the Blaschke product invariant subspaces for a composition operator are described. We show that if $\varphi$ has Denjoy-Wolff point $a$ on the unit circle, the atomic inner function subspaces with a single atom at $a$ are invariant subspaces for the composition operator $C_{\varphi }$.

References

Arne Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 239–255. MR 27954, DOI 10.1007/BF02395019
Carl C. Cowen, Iteration and the solution of functional equations for functions analytic in the unit disk, Trans. Amer. Math. Soc. 265 (1981), no. 1, 69–95. MR 607108, DOI 10.1090/S0002-9947-1981-0607108-9
Carl C. Cowen, Composition operators on $H^{2}$, J. Operator Theory 9 (1983), no. 1, 77–106. MR 695941
Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. MR 1397026
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
M. M. Jones, Shift invariant subspaces of composition operators on $H^p$, Arch. Math. (Basel) 84 (2005), no. 3, 258–267. MR 2134140, DOI 10.1007/s00013-004-1017-z
Ali Mahvidi, Invariant subspaces of composition operators, J. Operator Theory 46 (2001), no. 3, suppl., 453–476. MR 1897149
D. J. Newman, Interpolation in $H^{\infty }$, Trans. Amer. Math. Soc. 92 (1959), 501–507. MR 117350, DOI 10.1090/S0002-9947-1959-0117350-X
H. S. Shapiro and A. L. Shields, On some interpolation problems for analytic functions, Amer. J. Math. 83 (1961), 513–532. MR 133446, DOI 10.2307/2372892