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.

The Transcriptional Regulation of Retrotransposon BARE

The purpose of this research project was to understand the steps of the retrotransposon BARE (BArley REtrotransposon) life cycle, from regulation of transcription to Virus-Like Particle (VLP) formation and ultimate integration back into the genome. Our study concentrates mainly on BARE1 transcriptional regulation because transcription is the crucial first step in the retrotransposon life cycle. The BARE element is a Class I LTR (Long Terminal Repeat) retrotransposon belonging to the Copia superfamily and was originally isolated in our research group. The LTR retrotransposons are transcribed from promoters in the LTRs and encode proteins for packaging of their transcripts, the reverse transcription of the transcripts into cDNA, and integration of the cDNA back into the genome. BARE1 is translated as a single polyprotein and cleaved into the capsid protein (GAG), integrase (IN), and reverse transcriptase-RNaseH (RT-RH) by the integral aspartic proteinase (AP). The BARE retrotransposon family comprises more than 104 copies in the barley (Hordeum vulgare) genome. The element is bound by long terminal repeats (LTRs, 1829 bp) containing promoters required for replication, signals for RNA processing, and motifs necessary for the integration of the cDNA. Members of the BARE1 subfamily are transcribed, translated, and form virus-like particles. Several basic questions concerning transcription are explored in the thesis: BARE1 transcription control, promoter choice in different barley tissues, start and termination sites for BARE transcripts, and BARE1 transcript polyadenylation (I). Polyadenylation is an important step during mRNA maturation, and determines its stability and translatability among other characteristics. Our work has found a novel way used by BARE1 to make extra GAG protein, which is critical for VLP formation. The discovery that BARE1 uses one RNA population for protein synthesis and another RNA population for making cDNA has established the most important step of the BARE1 life cycle (III). The relationship between BARE1 and BARE2 has been investigated. Besides BARE, we have examined the retrotransposon Cassandra (II), which uses a very different transcriptional mechanism and a fully parasitic life cycle. In general, this work is focused on BARE1 promoter activity, transcriptional regulation including differential promoter usage and RNA pools, extra GAG protein production and VLP formation. The results of this study give new insights into transcription regulation of LTR retrotransposons.