Hilbert spaces of analytic functions between the Hardy and the Dirichlet space

Author:
Alexandru Aleman

Journal:
Proc. Amer. Math. Soc. **115** (1992), 97-104

MSC:
Primary 46E20; Secondary 30D50

MathSciNet review:
1079693

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For a large class of Hilbert spaces of analytic functions in the unit disc lying between the Hardy and the Dirichlet space we prove that each element of the space is the quotient of two bounded functions in the same space. It follows that the multiplication operator on these spaces is cellular indecomposable and that each invariant subspace contains nontrivial bounded functions.

**[1]**Leon Brown and Allen L. Shields,*Cyclic vectors in the Dirichlet space*, Trans. Amer. Math. Soc.**285**(1984), no. 1, 269–303. MR**748841**, 10.1090/S0002-9947-1984-0748841-0**[2]**Stefan Richter,*Invariant subspaces in Banach spaces of analytic functions*, Trans. Amer. Math. Soc.**304**(1987), no. 2, 585–616. MR**911086**, 10.1090/S0002-9947-1987-0911086-8**[3]**Stefan Richter and Allen Shields,*Bounded analytic functions in the Dirichlet space*, Math. Z.**198**(1988), no. 2, 151–159. MR**939532**, 10.1007/BF01163287**[4]**Joel H. Shapiro,*The essential norm of a composition operator*, Ann. of Math. (2)**125**(1987), no. 2, 375–404. MR**881273**, 10.2307/1971314

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
46E20,
30D50

Retrieve articles in all journals with MSC: 46E20, 30D50

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1992-1079693-X

Article copyright:
© Copyright 1992
American Mathematical Society