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.

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 |