Some Hilbert spaces of analytic functions. I

Abstract

The spaces of analytic functions studied arise from the problem of finding invariant subspaces for bounded linear transformations in Hilbert space. Two fundamental problems are (1) to determine the invariant subspaces of any bounded transformation, (2) to reconstruct a transformation from its invariant subspaces. Satisfactory answers are known for self-adjoint transformations, for unitary transformations, and more generally for normal transformations, but for all other kinds of transformations the known results are less complete. Beurling [2] illustrates how to find invariant subspaces for isometric transformations which are not unitary, and in so doing uncovers an important connection with analytic function theory. Aronszajn and Smith [1] are able to show the existence of invariant subspaces for completely continuous transformations, a result which they ascribe to von Neumann in the Hilbert space case with which we are now concerned. A general study of transformations T with TT* completely continuous is started by Livsic [11] and continued by Brodskil and Livsic [9]. The Livsic approach forms an interesting link between the methods of analytic function theory and those which depend on compactness in linear spaces. When the spectrum of the transformation is a point, Gohberg and Krein [10] give an integral representation of the transformation in terms of invariant subspaces. This construction makes an interesting contrast with the spectral representation of a self-adjoint operator. Relations between the existence of invariant subspaces and the factorization of related operator valued entire functions are obtained by Brodskil' [7; 8]. Our purpose now is to give an exposition of the function theoretic background to this interesting observation. Recall that we have previously [3-6] made a study of Hilbert spaces, whose elements are entire functions and which have these properties: (H1) Whenever F(z) is in the space and has a nonreal zero w, the function F(z)(z w)/(z w) is in the space and has the same norm as F(z). (H2) For every complex number w, the linear functional defined on the space by F(z) -+ F(w) is continuous. (H3) Whenever F(z) is in the space, the function F*(z)=F(i) is in the space and has the same norm as F(z). The axiom (H2) which appears here is conjectured to be a consequence of (Hi). Several apparently weaker conditions, of various degrees of subtlety, are known to imply (H2), and one of these is quoted