Boundary Values of Cauchy Type Integrals

Results by A. G. Poltoratskĭı and A. B. Aleksandrov about nontangential boundary values of pseudocontinuable H2-functions on sets of zero Lebesgue measure are used for the study of operators on L2-spaces on the unit circle. For an arbitrary bounded operator X acting from one such L2-space to another and having the property that the commutator of it with multiplication by the independent variable is a rank one operator, it is shown that X can be represented as a sum of multiplication by a function and a Cauchy transformation in the sense of angular boundary values. This article is devoted to the nontangential, or angular, boundary values for certain classes of functions holomorphic in the unit disk of the complex plane. For functions in the Hardy classes H, the existence of angular boundary values almost everywhere with respect to Lebesgue measure is a classical fact. However, much less is known what happens if we replace the Lebesgue measure by a singular measure. In the paper [1] by Poltoratskĭı, a series of results in this context were obtained for functions that are ratios of two Cauchy transforms of complex measures on the unit circle. These results are closely related to the existence problem for boundary values of functions in subspaces Kθ of the Hardy space H (these are precisely the subspaces invariant under the backward shift operator). Actually, in [1] the existence of angular boundary values was proved for Cauchy transforms in the case of the spaces L(σα) for special measures σα linked with an inner function θ. These examples will be discussed after Theorem 1. Aleksandrov [2] proved (see Theorem 2 below) that if the operator Kθ → L(μ), taking continuous functions in Kθ to their boundary values, is continuous, then nontangential boundary values exist μ-almost everywhere for all functions in Kθ (here μ is a measure on the unit circle, and L(μ) denotes the space of all measurable functions with the topology of convergence in μ-measure). Our main result has the same meaning: the boundedness of a Cauchy transformation yields the existence of angular boundary values. With the help of Aleksandrov’s result just mentioned, we shall describe the continuous operators that act from Kθ to the space of μ-measurable functions on the circle and “almost commute” with multiplication by the independent variable. Our main theorem can also be viewed in the framework of scattering theory: in a simple (scalar) case, we show that a wave operator exists also without any assumptions (which are present in the classical results) about the absolute continuity of the spectrum, i.e., we admit that the spectral measure of a unitary operator under consideration may have a singular part. For Cauchy type integrals we use the notation Kα(λ) = ∫ dα(z) 1− z̄λ , 2000 Mathematics Subject Classification. Primary 30E20, 47B47.