Functions Which Are Almost Multipliers of Hilbert Function Spaces

We introduce a natural class of functions, the pseudomultipliers, associated with a general Hilbert function space, prove an extension theorem which justifies the definition, give numerous examples and establish the nature of the 1-pseudomultipliers of Hilbert spaces of analytic functions under mild hypotheses. The function 1/z on the unit disc D is almost a multiplier of the Hardy space H: it misses by only one dimension. That is, there is a closed subspace of H of codimension 1 which is multiplied by 1/z into H. The same statement holds for the characteristic function of the point 0. These are two key examples of functions that we call pseudomultipliers of Hilbert function spaces. Now multipliers of the standard function spaces have been much studied: it is a natural generalization to consider functions which fail to be multipliers by only finitely many dimensions. Moreover, for some familiar function spaces one obtains in this way natural classes of functions. For example, a well-known theorem of Adamyan, Arov and Krein [AAK] on s-numbers of Hankel operators can be interpreted as a description of the pseudomultipliers of H (see Theorem 3.4 below). They are the finite modifications of functions of the form f/p where f is bounded and analytic in the unit disc and p is a polynomial which does not vanish on the unit circle. The pseudomultipliers of the Fock space (see Theorem 3.5 below) are the finite modifications of the proper rational functions. We address the question of what can be said about the pseudomultipliers of other popular function spaces. One can formulate the definition of a pseudomultiplier of a function space in very great generality. In this paper we introduce the notion for the case of Hilbert function spaces. This still covers a very wide variety of spaces, but we are nevertheless able to make significant assertions about them. A major purpose of the paper is to give the Date: May 23, 1996. 1991 Mathematics Subject Classification. AMS Subject Classifications: 46E22, 46E20.