Some new results concerning Schur multipliers and duality results between Bergman-Schatten and little Bloch spaces

This Licentiate thesis consists of an introduction and three papers, which deal with some spaces of infinite matrices. In the introduction we give an overview of the area that serves as a frame for the rest of the thesis.In Paper 1 we introduce the space $B_w(\ell^2)$ of linear (unbounded) operators on $\ell^2$ which map decreasing sequences from $\ell^2$ into sequences from $\ell^2$ and we find some classes of operators belonging either to $B_w(\ell^2)$ or to the space of all Schur multipliers on $B_w(\ell^2)$.In Paper 2 we further continue the study of the space $B_w(\ell^p)$ in the range $1 <\infty$. In particular, we characterize the upper triangular positive matrices from $B_w(\ell^p)$.In Paper 3 we prove a new characterization of the Bergman-Schatten spaces $L_a^p(D,\ell^2)$, the space of all upper triangular matrices such that $\|A(\cdot)\|_{L^p(D,\ell^2)}<\infty$, where \[\|A(r)\|_{L^p(D,\ell^2)}=\left(2\int_0^1\|A(r)\|^p_{C_p}rdr\right)^\frac{1}{p}. \]This characterization is similar to that for the classical Bergman spaces. We also prove a duality between the little Bloch space and the Bergman-Schatten classes in the case of infinite matrices.Godkänd; 2009; 20090423 (livmar); LICENTIATSEMINARIUM Ämnesområde: Matematik/Mathematics Examinator: Professor Lars-Erik Persson, Luleå tekniska universitet Tid: Tisdag den 2 juni 2009 kl 10.15 Plats: D 2214, Luleå tekniska universite