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The classical Korenblum-Roberts Theorem characterizes the cyclic singular inner functions in the Bergman spaces of the unit disk D as those for which the corresponding singular measure vanishes on Beurling-Carleson sets of Lebesgue measure zero. We solve the weighted variant of the problem in which the Bergman space is replaced by a Pt(mu) space, the closure of analytic polynomials in a Lebesgue space & Laplacetrf;t(mu) corresponding to a measure of the form dA alpha + w dm, with dA alpha being the standard weighted area measure on D, dm the Lebesgue measure on the unit circle T, and w a general weight on T. We characterize when Pt(mu) of this form is a space of analytic functions on D by computing the Thomson decomposition of the measure mu. The structure of the decomposition is expressed in terms of what we call the family of associated Beurling-Carleson sets. We characterize the cyclic singular inner functions in the analytic Pt(mu) spaces as those for which the corresponding singular measure vanishes on the family of associated Beurling-Carleson sets. Unlike the classical setting, Beurling-Carleson sets of both zero and positive Lebesgue measure appear in our description. As an application of our results, we complete the characterization of the symbols b : D -> D which generate a de Branges-Rovnyak space with a dense subset of functions smooth on T. The characterization is given explicitly in terms of the modulus of bon T and the singular measure corresponding to the singular inner factor of b. Our proofs involve Khrushchev's techniques of simultaneous polynomial approximations and linear programming ideas of Korenblum, combined with recently established constrained C1-optimization tools.

Given a compact convex planar domain Omega with non-empty interior, the classical Neumann's configuration constant c(R)(Omega) is the norm of the Neumann-Poincar & eacute; operator K Omega acting on the space of continuous real-valued functions on the boundary partial derivative Omega, modulo constants. We investigate the related operator norm cC(Omega) of K Omega on the corresponding space of complex-valued functions, and the norm a(Omega) on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality c(R)(Omega)=cC(Omega), the analytic Neumann-type inequality a(Omega)<1, and provide various estimates for these quantities expressed in terms of the geometry of Omega. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type parallel to p(T)parallel to <= Ksup(z is an element of Omega)|p(z)|, where p is a polynomial and Omega is a domain containing the numerical range of the operator T. Among other results, we show that the well-known Crouzeix-Palencia bound K <= 1+root 2- can be improved to K <= 1+root 1+a(Omega). In the case that Omega is an ellipse, this leads to an estimate of K in terms of the eccentricity of the ellipse.

We revisit the problem of characterizing cyclic elements for the shift operator in a broad class of radial growth spaces of holomorphic functions on the unit disk, focusing on functions of finite Nevanlinna characteristic. We provide results in the range of Dini regular weights, and in the regime of logarithmic integral divergence. Our proofs are largely constructive and allow for substantial simplifications of earlier works that previously relied on the Carleson Corona Theorem, such as the Korenblum-Roberts Theorem, as well as a more recent result of El-Fallah, Kellay and Seip.

For a positive finite Borel measure mu compactly supported in the complex plane, the space P2(mu) is the closure of the analytic polynomials in the Lebesgue space L2(mu). According to Thomson's famous result, any space P2(mu) decomposes as an orthogonal sum of pieces which are essentially analytic, and a residual L2-space. We study the structure of this decomposition for a class of Borel measures mu supported on the closed unit disk D for which the part mu D, living in the open disk D, is radial and decreases at least exponentially fast near the boundary of the disk. For the considered class of measures, we give a precise form of the Thomson decompsition. In particular, we confirm a conjecture of Kriete and MacCluer from 1990, which gives an analog to Szeg & ouml;'s classical theorem.

We study conditions for the containment of a given space X of analytic functions on the unit disk D in the de Branges-Rovnyak space 9L(b). We deal with the nonextreme case in which b admits a Pythagorean mate a, and derive a multiplier boundedness criterion on the function 0 = b/a which implies the containment X C 9L(b). With our criterion, we are able to characterize the containment of the Hardy space 9Lp inside 9L(b) for p E [2, oo]. The end-point cases have previously been considered by Sarason, and we show that in his result, stating that 0 E 9L2 is equivalent to 9L infinity C 9L(b), one can in fact replace 9L infinity by BMOA. We establish various other containment results, and study in particular the case of the Dirichlet space D, whose containment is characterized by a Carleson measure condition. In this context, we show that matters are not as simple as in the case of the Hardy spaces, and we carefully work out an example.

For a given Beurling–Carleson subset E of the unit circle T which has positive Lebesgue measure, we give explicit formulas for measurable functions supported on E such that their Cauchy transforms have smooth extensions from D to T. The existence of such functions has been previously established by Khrushchev in 1978, in non-constructive ways by the use of duality arguments. We construct several families of such smooth Cauchy transforms and apply them in a few related problems in analysis: an irreducibility problem for the shift operator, an inner factor permanence problem. Our development leads to a self-contained duality proof of the density of smooth functions in a very large class of de Branges–Rovnyak spaces. This extends the previously known approximation results.

For the class of de Branges-Rovnyak spaces  of the unit disk  defined by extreme points b of the unit ball of , we study the problem of approximation of a general function in  by a function with an extension to the unit circle  of some degree of smoothness, for instance satisfying Hölder estimates or being differentiable. We will exhibit connections between this question and the theory of subnormal operators and, in particular, we will tie the possibility of smooth approximations to properties of invariant subspaces of a certain subnormal operator. This leads us to several computable conditions on b which are necessary for such approximations to be possible. For a large class of extreme points b we use our result to obtain explicit necessary and sufficient conditions on the symbol b which guarantee the density of functions with differentiable boundary values in the space . These conditions include an interplay between the modulus of b on  and the spectrum of its inner factor.

We study the classical problem of identifying the structure of , the closure of analytic polynomials in the Lebesgue space of a compactly supported Borel measure living in the complex plane. In his influential work, Thomson [Ann. of Math. (2) 133 (1991), pp. 477–507] showed that the space decomposes into a full -space and other pieces which are essentially spaces of analytic functions on domains in the plane. For a family of measures supported on the closed unit diskwhich have a part on the open disk which is similar to the Lebesgue area measure, and a part on the unit circle which is the restriction of the Lebesgue linear measure to a general measurable subset of , we extend the ideas of Khrushchev and calculate the exact form of the Thomson decomposition of the space . It turns out that the space splits according to a certain natural decomposition of measurable subsets of which we introduce. We highlight applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces. 

We provide an abstract approach to approximation with a wide range of regularity classes X in spaces of pseudocontinuable functions Kp Θ, where Θ is an inner function and p > 0. More precisely, we demonstrate a general principle, attributed to A. Aleksandrov, which asserts that if a certain linear manifold X is dense in Kq Θ for some q > 0, then X is in fact dense in Kp Θ for all p > 0. Moreover, for a rich class of Banach spaces of analytic functions X, we describe the precise mechanism that determines when X is dense in a certain space of pseudocontinuable functions. As a consequence, we obtain an extension of Aleksandrov's density theorem to the class of analytic functions with uniformly convergent Taylor series. © 2022 American Mathematical Society.

It is well known that for any inner function θ defined in the unit disk D, the following two conditions: (i) there exists a sequence of polynomials {pn}n such that limn→∞θ(z)pn(z)=1 for all z∈D and (ii) supn∥θpn∥∞<∞, are incompatible, i.e., cannot be satisfied simultaneously. However, it is also known that if we relax the second condition to allow for arbitrarily slow growth of the sequence {θ(z)pn(z)}n as |z|→1, then condition (i) can be met for some singular inner function. We discuss certain consequences of this fact which are related to the rate of decay of Taylor coefficients and moduli of continuity of functions in model spaces Kθ. In particular, we establish a variant of a result of Khavinson and Dyakonov on nonexistence of functions with certain smoothness properties in Kθ, and we show that the classical Aleksandrov theorem on density of continuous functions in Kθ is essentially optimal. We consider also the same questions in the context of de Branges–Rovnyak spaces H(b) and show that the corresponding approximation result also is optimal.