Composition operators on vector-valued BMOA and related function spaces

A composition operator is a linear operator between spaces of analytic or harmonic functions on the unit disk, which precomposes a function with a fixed self-map of the disk. A fundamental problem is to relate properties of a composition operator to the function-theoretic properties of the self-map. During the recent decades these operators have been very actively studied in connection with various function spaces. The study of composition operators lies in the intersection of two central fields of mathematical analysis; function theory and operator theory. This thesis consists of four research articles and an overview. In the first three articles the weak compactness of composition operators is studied on certain vector-valued function spaces. A vector-valued function takes its values in some complex Banach space. In the first and third article sufficient conditions are given for a composition operator to be weakly compact on different versions of vector-valued BMOA spaces. In the second article characterizations are given for the weak compactness of a composition operator on harmonic Hardy spaces and spaces of Cauchy transforms, provided the functions take values in a reflexive Banach space. Composition operators are also considered on certain weak versions of the above function spaces. In addition, the relationship of different vector-valued function spaces is analyzed. In the fourth article weighted composition operators are studied on the scalar-valued BMOA space and its subspace VMOA. A weighted composition operator is obtained by first applying a composition operator and then a pointwise multiplier. A complete characterization is given for the boundedness and compactness of a weighted composition operator on BMOA and VMOA. Moreover, the essential norm of a weighted composition operator on VMOA is estimated. These results generalize many previously known results about composition operators and pointwise multipliers on these spaces.Väitöskirjassa tutkitaan kompositio-operaattoreita kompleksitason yksikkökiekon analyyttisten ja harmonisten funktioiden avaruuksissa. Kompositio-operaattori on lineaarinen kuvaus, joka yhdistää funktioon sisältä päin symbolin eli jonkin analyyttisen kuvauksen yksikkökiekolta itselleen. Tutkimuksen tavoitteena on kuvata kompositio-operaattorin ominaisuuksia symbolin funktioteoreettisten ominaisuuksien avulla. Viimeisten vuosikymmenien aikana kompositio-operaattoreita on tutkittu aktiivisesti eri funktioavaruuksissa. Tutkimus sijoittuu kahden keskeisen matemaattisen analyysin osa-alueen, funktioteorian ja operaattoriteorian leikkaukseen. Väitöskirja koostuu neljästä artikkelista ja yhteenveto-osasta. Ensimmäisessä ja kolmannessa artikkelissa annetaan riittäviä ehtoja kompositio-operaattorin heikolle kompaktisuudelle vektoriarvoisissa BMOA-avaruuksissa ja sen eri versioissa. Toisessa artikkelissa karakterisoidaan operaattorin heikko kompaktisuus vektoriarvoisissa harmonisissa Hardy-avaruuksissa ja Cauchy-muunnosten avaruuksissa. Kompositio-operaattoreita tutkitaan myös näiden avaruuksien heikoissa versioissa. Lisäksi eri avaruuksien välisiä eroja valaistaan esimerkein. Viimeisessä artikkelissa tutkitaan painotettuja kompositio-operaattoreita skalaariarvoisessa BMOA-avaruudessa ja sen aliavaruudessa VMOA. Painotettu kompositio-operaattori saadaan soveltamalla ensin kompositio-operaattoria ja sitten pisteittäistä multiplikaattoria. Työssä karakterisoidaan painotetun kompositio-operaattorin kompaktisuus BMOA-avaruudessa ja annetaan arvio operaattorin olennaiselle normille VMOA-avaruudessa. Nämä tulokset yleistävät monia aikaisemmin tunnettuja kompositio-operaattoreita ja multiplikaattoreita koskevia tuloksia

Convexity and the Euclidean metric of space-time

We address the question about the reasons why the "Wick-rotated", positive-definite, space-time metric obeys the Pythagorean theorem. An answer is proposed based on the convexity and smoothness properties of the functional spaces purporting to provide the kinematic framework of approaches to quantum gravity. We employ moduli of convexity and smoothness which are eventually extremized by Hilbert spaces. We point out the potential physical significance that functional analytical dualities play in this framework. Following the spirit of the variational principles employed in classical and quantum Physics, such Hilbert spaces dominate in a generalized functional integral approach. The metric of space-time is induced by the inner product of such Hilbert spaces.Comment: 41 pages. No figures. Standard LaTeX2e. Change of affiliation of the author and mostly superficial changes in this version. Accepted for publication by "Universe" in a Special Issue with title: "100 years of Chronogeometrodynamics: the Status of Einstein's theory of Gravitation in its Centennial Year

On the local solvability of the initial-boundary value problem of fiber spinning of the upper convected Maxwell fluid

The fiber spinning process of a viscoelastic liquid modeled by the constitutive theory of the Maxwell fluid is analyzed.The governing equations are given by one- dimensional mass, momentum, and constitutive equations which arise in the slender bodyapproximation by cross-sectional averaging of the two-dimensional axisymmetric Stokes equationswith free boundary. Existence, uniqueness, andregularity results are proved by means of fixed point arguments, energy estimates, and weak/weak* convergence methods.The complexity in this problem lies with the constitutive model of the Maxwell fluid: when both the outflow velocity at the spinneret andthe pulling velocity at take-upare prescribed, a boundary condition can be imposed for only one of the two elastic stress components at the inlet. The absence ofthe second stress boundary condition makes the mathematical analysis of the problem difficult