Composition operators, Aleksandrov measures and value distribution of analytic maps in the unit disc

A composition operator is a linear operator that precomposes any given function with another function, which is held fixed and called the symbol of the composition operator. This dissertation studies such operators and questions related to their theory in the case when the functions to be composed are analytic in the unit disc of the complex plane. Thus the subject of the dissertation lies at the intersection of analytic function theory and operator theory. The work contains three research articles. The first article is concerned with the value distribution of analytic functions. In the literature there are two different conditions which characterize when a composition operator is compact on the Hardy spaces of the unit disc. One condition is in terms of the classical Nevanlinna counting function, defined inside the disc, and the other condition involves a family of certain measures called the Aleksandrov (or Clark) measures and supported on the boundary of the disc. The article explains the connection between these two approaches from a function-theoretic point of view. It is shown that the Aleksandrov measures can be interpreted as kinds of boundary limits of the Nevanlinna counting function as one approaches the boundary from within the disc. The other two articles investigate the compactness properties of the difference of two composition operators, which is beneficial for understanding the structure of the set of all composition operators. The second article considers this question on the Hardy and related spaces of the disc, and employs Aleksandrov measures as its main tool. The results obtained generalize those existing for the case of a single composition operator. However, there are some peculiarities which do not occur in the theory of a single operator. The third article studies the compactness of the difference operator on the Bloch and Lipschitz spaces, improving and extending results given in the previous literature. Moreover, in this connection one obtains a general result which characterizes the compactness and weak compactness of the difference of two weighted composition operators on certain weighted Hardy-type spaces.

Effects of unilateral vocal fold paralysis treatment with fascia augmentation on airflow mechanics, tracheal sounds and voice acoustics

Data on the influence of unilateral vocal fold paralysis on breathing, especially other than information obtained by spirometry, are relatively scarce. Even less is known about the effect of its treatment by vocal fold medialization. Consequently, there was a need to study the issue by combining multiple instruments capable of assessing airflow dynamics and voice. This need was emphasized by a recently developed medialization technique, autologous fascia injection; its effects on breathing have not previously been investigated. A cohort of ten patients with unilateral vocal fold paralysis was studied before and after autologous fascia injection by using flow-volume spirometry, body plethysmography and acoustic analysis of breathing and voice. Preoperative results were compared with those of ten healthy controls. A second cohort of 11 subjects with unilateral vocal fold paralysis was studied pre- and postoperatively by using flow-volume spirometry, impulse oscillometry, acoustic analysis of voice, voice handicap index and subjective assessment of dyspnoea. Preoperative peak inspiratory flow and specific airway conductance were significantly lower and airway resistance was significantly higher in the patients than in the healthy controls (78% vs. 107%, 73% vs. 116% and 182% vs. 125% of predicted; p = 0.004, p = 0.004 and p = 0.026, respectively). Patients had a higher root mean square of spectral power of tracheal sounds than controls, and three of them had wheezes as opposed to no wheezing in healthy subjects. Autologous fascia injection significantly improved acoustic parameters of the voice in both cohorts and voice handicap index in the latter cohort, indicating that this procedure successfully improved voice in unilateral vocal fold paralysis. Peak inspiratory flow decreased significantly as a consequence of this procedure (from 4.54 ± 1.68 l to 4.21 ± 1.26 l, p = 0.03, in pooled data of both cohorts), but no change occurred in the other variables of flow-volume spirometry, body-plethysmography and impulse oscillometry. Eight of the ten patients studied by acoustic analysis of breathing had wheezes after vocal fold medialization compared with only three patients before the procedure, and the numbers of wheezes per recorded inspirium and expirium increased significantly (from 0.02 to 0.42 and from 0.03 to 0.36; p = 0.028 and p = 0.043, respectively). In conclusion, unilateral vocal fold paralysis was observed to disturb forced breathing and also to cause some signs of disturbed tidal breathing. Findings of flow volume spirometry were consistent with variable extra-thoracic obstruction. Vocal fold medialization by autologous fascia injection improved the quality of the voice in patients with unilateral vocal fold paralysis, but also decreased peak inspiratory flow and induced wheezing during tidal breathing. However, these airflow changes did not appear to cause significant symptoms in patients.

Properties of Toeplitz operators on analytic function spaces : from function symbols to distributions

Toeplitz operators are among the most important classes of concrete operators with applications to several branches of pure and applied mathematics. This doctoral thesis deals with Toeplitz operators on analytic Bergman, Bloch and Fock spaces. Usually, a Toeplitz operator is a composition of multiplication by a function and a suitable projection. The present work deals with generalizing the notion to the case where the function is replaced by a distributional symbol. Fredholm theory for Toeplitz operators with matrix-valued symbols is also considered. The subject of this thesis belongs to the areas of complex analysis, functional analysis and operator theory. This work contains five research articles. The articles one, three and four deal with finding suitable distributional classes in Bergman, Fock and Bloch spaces, respectively. In each case the symbol class to be considered turns out to be a certain weighted Sobolev-type space of distributions. The Bergman space setting is the most straightforward. When dealing with Fock spaces, some difficulties arise due to unboundedness of the complex plane and the properties of the Gaussian measure in the definition. In the Bloch-type spaces an additional logarithmic weight must be introduced. Sufficient conditions for boundedness and compactness are derived. The article two contains a portion showing that under additional assumptions, the condition for Bergman spaces is also necessary. The fifth article deals with Fredholm theory for Toeplitz operators having matrix-valued symbols. The essential spectra and index theorems are obtained with the help of Hardy space factorization and the Berezin transform, for instance. The article two also has a part dealing with matrix-valued symbols in a non-reflexive Bergman space, in which case a condition on the oscillation of the symbol (a logarithmic VMO-condition) must be added.

Paradoxical reflection in quantum mechanics

This article concerns a phenomenon of elementary quantum mechanics that is quite counter-intuitive, very non-classical, and apparently not widely known: a quantum particle can get reflected at a downward potential step. In contrast, classical particles get reflected only at upward steps. The conditions for this effect are that the wave length is much greater than the width of the potential step and the kinetic energy of the particle is much smaller than the depth of the potential step. This phenomenon is suggested by non-normalizable solutions to the time-independent Schroedinger equation, and we present evidence, numerical and mathematical, that it is also indeed predicted by the time-dependent Schroedinger equation. Furthermore, this paradoxical reflection effect suggests, and we confirm mathematically, that a quantum particle can be trapped for a long time (though not forever) in a region surrounded by downward potential steps, that is, on a plateau.