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Cyclic vectors in Banach spaces of analytic functions. (English) Zbl 0604.46028
Operators and function theory, Proc. NATO Adv. Study Inst., Lancaster/Engl. 1984, NATO ASI Ser., Ser. C 153, 319-349 (1985).
[For the entire collection see Zbl 0563.00012.]
This article is an excellent survey of the theory of cyclic vectors in Banach spaces of analytic functions, a theory in whose development the author has played an active role. The article is based on three lectures given by the author at a conference 1984 in Lancaster/England.
The paper consists of three parts. Let E be a Banach space of analytic functions on a bounded region G. In the first chapter the problem of cyclic vectors, i.e. those functions f with the property that span $$\{z^ nf:n=0,1,2,...\}$$ is dense in E is discussed for spaces E fulfilling some general axioms. An obvious necessary condition for f to be cyclic is that f has no zeros on G. In the examples found so far this condition is not sufficient, one has to impose additional conditions on f. For example, in the Hardy space $$H^ 2$$ a necessary and sufficient condition is that f is an outer function (Beurling, 1949). In the general case various results are established and several questions are posed.
In the second chapter the author considers the Bergman space B of analytic functions. In this case a variety of results is known, the most interesting is the following: A necessary and sufficient condition for a singular inner function $$S_{\mu}$$ to be cyclic in B is that the associated positive singular measure $$\mu$$ put no mass on any Carleson set of the unit circle. In the third chapter the Dirichlet space D is considered which consists of all analytic functions that map the unit disc onto a Riemann surface of finite area. A cyclic vector f in D must be an outer function and it is conjectured that f is cyclic in D if and only if f is an outer function with the property that the boundary zero set $Z(f):=\{e^{it}:\lim_{r\to 1}f(re^{it})=0\}$ has logarithmic capacity zero. Most of the material of the article under review is taken from the fundamental paper of Leon Brown and the author [Trans. Amer. Math. Soc. 285, 269-304 (1984; Zbl 0517.30040)]. The many open problems show that the field is far from being exhausted and much remains to be done here.
Reviewer: M.von Renteln

##### MSC:
 46E15 Banach spaces of continuous, differentiable or analytic functions 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 30H05 Spaces of bounded analytic functions of one complex variable