Cyclic Vectors in the Dirichlet Space

Abstract

Abstract. We study the Hubert space of analytic functions with finite Dirichlet integral in the open unit disc. We try to identify the functions whose polynomial multiples are dense in this space. Theorems 1 and 2 confirm a special case of the following conjecture: if \f(z)\ > \g(z)\ at all points and if g is cyclic, then/is cyclic. Theorems 3-5 give a sufficient condition (/ is an outer function with some smoothness and the boundary zero set is at most countable) and a necessary condition (the radial limit can vanish only for a set of logarithmic capacity zero) for a function / to be cyclic.Introduction. In this paper we shall study the (Hubert) space of analytic functions in the open unit disc A in the complex plane that have a finite Dirichlet integral: // l/'l2 dx dy < oo. Our goal is to identify, as far as possible, the "cyclic vectors" in this space, that is, those functions/such that the polynomial multiples of fare dense in the space. The corresponding problem for the Hardy space H2 was solved by Beurling [5] in 1949: the cychc vectors are precisely the outer functions. The present paper is divided into four sections; it contains 20 propositions, 5 theorems and 19 unsolved problems (stated as Questions in the text).The first section deals with cyclic vectors and multiplication operators in a general Banach space of analytic functions (in a bounded region of the complex plane). The theory is illustrated by considering a special family of Hubert spaces, denoted {Da}, -oo < a < oo, in the unit disc. (The values a = 0, 1 give, respectively, H2 and the Dirichlet space.) These spaces (for 0 < a < 1) were considered by Carleson in his dissertation [7]. This section contains 10 propositions and raises 6 questions, mostly for a general Banach space of analytic functions. For example, Question 3 asks if / must be cychc whenever we have \f(z)\> \g(z)\ for some cychc g, and all zl Question 4 asks if / must be cychc whenever / and 1// are both in the space. No examples are known where either of these questions has a negative answer.In §2 we begin the study of cychc vectors in the Dirichlet space D. Theorems 1 and 2 give a partial answer to Question 3 above, for this space. This section also contains 2 propositions (10,11) and 4 questions (7-10). Proposition 11 says that if/ and g are bounded functions in D whose product is cychc, then both / and g must be cychc. Theorem 2 gives a partial converse (we require that \g\ be Dini continuous on

Invariant Subspaces in Banach Spaces of Analytic Functions

“…It follows that h e Jt\ V Jt2, as we had to show. D Finally, that (3.11) is equivalent to (3.10) if 38 is reflexive follows from Proposition 2, p. 272 of [9]. Actually, there the authors assumed that 36 is separable, an assumption that follows automatically for reflexive Banach spaces of analytic functions satisfying axioms (1.1) and (1.2).…”

“…The corresponding norm will be denoted by || ||a. A short survey of the known results about Da is given in [9]. For a < 0 \\f\\a is equivalent to ||/||2,w with w{z) = {1 -|z|2)^1_a (see Example 2.8 and [27]), thus in this case Da = L2(D,(1 -|2|2)~1_aí¿4) with equivalence of norms.…”

Some Examples of Cyclic Vectors in the Dirichlet Space

“…For E, a closed subset of the unit circle with logarithmic capacity zero, we construct a function in this space which is uniformly continuous, vanishes on E, and is cyclic with respect to the shift operator. Question 12 of [1] asks if the converse of Theorem A holds for an outer function f € D. That is, letting Z(f) denote the zero set of /* for an outer function /, if Z(f*) has zero logarithmic capacity, must / be cyclic? In fact, no examples were given in [1] of cyclic vectors / for which Z(f) was uncountable.…”

“…Question 12 of [1] asks if the converse of Theorem A holds for an outer function f € D. That is, letting Z(f) denote the zero set of /* for an outer function /, if Z(f*) has zero logarithmic capacity, must / be cyclic? In fact, no examples were given in [1] of cyclic vectors / for which Z(f) was uncountable.…”

“…(2) If Da, 0 < a < 1, is the Hubert space of analytic functions for which ll/ll2 = Y2n°=o(n + l)al/(n)l2 's finite, then by using a suitable Bessel capacity instead of logarithmic capacity, one can prove theorems for Da corresponding to [1,Theorem 5] and Theorem B of this note.…”