Allen L. Shields

Cyclic vectors in the Dirichlet space

We study the Hilbert space of analytic functions with finite Dinchlet 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 IJ(z)I > Ig(z)l at all points and if g is cyclic, thenJis cyclic. Theorems 3-5 give a sufficient condition (t 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 loganthmic capacity zero) for a function J to be cyclic. Introduction. In this paper we shall study the (Hilbert) space of analytic functions in the open unit disc a in the complex plane that have a finite Dirichlet integral: JJ If t12 dx dy [g(z)l for some cyclic g, and all z? Question 4 asks if f must be cyclic whenever f and l/f 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 cyclic 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 f and g are bounded functions in D whose product is cyclic, then bothf and g must be cyclic. Theorem 2 gives a partial converse (we require that [gl be Dini continuous on Received by the editors July 21, 1983. 1980 Mathematics Subject Classification. Primary 30H05; Secondary 46E15, 46E20, 47B37.

Almost commuting matrices

It is shown that if A and B are n × n complex matrices with A = A∗ and ∥AB − BA∥</ 2ϵ2(n − 1), then there exist n × n matrices A′ and B′ with A′ = A′∗ such that A′B′ = B′A′ and ∥A − A′∥⩽ ϵ, ∥B − B′∥⩽ ϵ.

Mergelyan sets and the modulus of continuity of analytic functions

For f a function continuous on the closed unit disk and analytic in the interior, let 4% f) = sup{1 f(z1) f(zz)l : I Zl zg I < 6, I Zl I < 1, I -3 I B 1) and wt f> = sup{1 f(z1) f(zz>l : I Zl z2 I < 6, I z1 I = I z2 I = l> denote, respectively, the modulus of continuity off on the closed unit disk and the modulus of continuity of the restriction offto the boundary { 1 z I= l}. We consider here the question of determining the relationship of w(S,f) and G(S, f). Clearly, one has &(S, f) < ~(6, f), and we are concerned here with the extent to which the reverse inequality holds. For certain measures of growth, &(S,f) and w(S,f) are the same. For example, if 01 is given (0 < 01 < 1) and if &(S,f) < S”, then w(S,f) < Sm (see Theorem 2.2). However, it is not true in general that ~(6, f) = &(S, f) (see Section 4 for an example). Nevertheless, we do have the following result. The disk algebra A denotes the class of functions f that are continuous onIzI<landanalyticinIzI<l.

Cyclic Vectors in Banach Spaces of Analytic Functions

In these three lectures we consider Banach spaces of analytic functions on plane domains. If the space admits the operator of multiplication by z, then it is of interest to describe the cyclic vectors for this operator, that is, those functions in the space with the property that the polynomial multiples of the function are dense. A necessary condition is that the function have no zeros; in general it is difficult to give necessary and sufficient conditions.

On Badly Approximable Functions

Received November 2, 1974 1. INTRODUCTION Let D be a bounded domain in the complex plane with boundary r, and let A(D) be the algebra of analytic functions on D which extend continuously to r. The distance from a function v E C(r) to A(D) is defined to be 4~ A(D)) = W F -fll :f~ A(D)), where the norm is the supremum norm over r. In this paper, we consider the problem of describing the functions 9) E C(I’) which satisfy Such functions, excepting the function 0, will be called

Extreme Points in Quotients of Operator Algebras

Let A be a nest algebra and K the ideal of compact operators in L(H). We ask whether or not the closed unit ball of \(\frac{{L(H)}} {{A + K}}\) has any extreme points and find that the answer depends on the structure of the nest involved. For nests with order type of the extended integers and finite dimensional atoms, we completely characterize the extreme points and show that the closed convex hull of these is not all of Ball \(\left( {\frac{{L(H)}} {{A + K}}} \right)\).