https://www.mdu.se/

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.

We characterize the model spaces KΘ​ in which functions with smooth boundary extensions are dense. It is shown that such approximations are possible if and only if the singular measure associated to the singular inner factor of Θ is concentrated on a countable union of Beurling–Carleson sets. In fact, we use a duality argument to show that if there exists a restriction of the associated singular measure which does not assign positive measure to any Beurling–Carleson set, then even larger classes of functions, such as Hölder classes and large collections of analytic Sobolev spaces, fail to be dense. In contrast to earlier results on density of functions with continuous extensions to the boundary in KΘ​ and related spaces, the existence of a smooth approximant is obtained through a constructive method.

For the class of Hardy spaces and standard weighted Bergman spaces of the unit disk, we prove that the spectrum of a generalized Cesàro operator  is unchanged if the symbol  is perturbed to  by an analytic function  inducing a quasi-nilpotent operator ⁠, that is, spectrum of  equals ⁠. We also show that any  operator that can be approximated in the operator norm by an operator  with bounded symbol  is quasi-nilpotent. In the converse direction, we establish an equivalent condition for the function  to be in the BMOA norm closure of ⁠. This condition turns out to be equivalent to quasi-nilpotency of the operator  on the Hardy spaces. This raises the question whether similar statement is true in the context of Bergman spaces and the Bloch space. Furthermore, we provide some general geometric properties of the spectrum of  operators.

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift  is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case.

We study the spectrum of generalized Cesàro operators  acting on the class of growth spaces . We show how the problem of determining the spectrum is related to boundedness of standard weighted Bergman projections on weighted -spaces. Using this relation we establish some general spectral properties of these operators, and explicitly compute the spectrum for a large class of symbols g.