Composition Operators on Weighted Bergman and S^P Spaces

Let $varphi$ be an analytic self-map of open unit disk $mathbb{D}$. The operator given by $(C_{varphi}f)(z)=f(varphi(z))$, for $z in mathbb{D}$ and  $f$ analytic on $mathbb{D}$ is called composition operator. For each $pgeq 1$, let $S^p$ be the space of analytic functions on $mathbb{D}$ whose derivatives belong to the Hardy space $H^p$.  For alpha > -1 and p > 0 the weighted Bergman space $A^{p}_{alpha}$ consists of all analytic functions in $L^{p}(mathbb{D}, dA_{alpha})$, where $dA_{alpha}$ is the normalized weighted area measure. In this presentation, we characterize boundedness and compactness of composition operators act between weighted Bergman $A_{alpha}^{p}$ and $S^q$ spaces, 1leq p, q<infty. Moreover, we give a lower bound for the essential norm of composition operator from  $A_{alpha}^{p}$ into $S^q$ spaces, $1leq pleq q$

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

Essential norms of weighted composition operators between Hardy spaces HpH^p and HqH^q for 1≤p,q≤∞1\leq p,q \leq \infty

We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^p$ and $H^q$ for $1\leq p,q\leq\infty.$ In particular we give some estimates for the cases $1=p\leq q\leq\infty$ and $1\leq q<p\leq\infty.$Comment: 16 page

Approximation numbers of weighted composition operators

We study the approximation numbers of weighted composition operators $f\mapsto w\cdot(f\circ\varphi)$ on the Hardy space $H^2$ on the unit disc. For general classes of such operators, upper and lower bounds on their approximation numbers are derived. For the special class of weighted lens map composition operators with specific weights, we show how much the weight $w$ can improve the decay rate of the approximation numbers, and give sharp upper and lower bounds. These examples are motivated from applications to the analysis of relative commutants of special inclusions of von Neumann algebras appearing in quantum field theory (Borchers triples).Comment: 35 pages, no figures. Some typos removed, minor improvements in presentation, updated reference