Analogues of a Theorem of Frostman on Linear Fractional Transformations of Inner Functions and the Typical Spectral Structure of Analytic Families of Weak Contractions

H. Frostman [3] proved in 1935 the following remarkable theorem: if x(ζ) is an inner function defined in the open unit disk $$\mathbb{D}$$ , then $$f(\zeta ,z) = \frac{{x(\zeta ) - z}}{{1 - z*x(\zeta )}}$$ is, as a function of ζ, a Blaschke product for all the values of the parameter $$z \in \mathbb{D}$$ , with a possible exception of a set of points M f of zero logarithmic capacity (cap0 M f = 0). In 1955 M. Heins [12] established the following: if x(ζ) is an arbitrary function holomorphic and strictly contractive in $$\mathbb{D}$$ , then the function f (ζ, z) defined by (0.1) is the product of a Blaschke product and of an outer function for all $$z \in \mathbb{D}\backslash {{M}_{f}}$$ (M f of zero logarithmic capacity).