A study of Schur multipliers and some Banach spaces of infinite matrices

Abstract: This PhD thesis consists of an introduction and five papers, which deal with some spaces of infinite matrices and Schur multipliers.
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
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(2int_0^1|A(r)|^p_{C_p}rdr ight)^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.
In Papers 4 we prove a duality result between $B_p(ell^2)$ and $B_q(ell^2)$, $1 [|A|_{B_p(ell^2)}=left[int_0^1 (1-r^2)^{2p}|A''(r)|_{C_p}^pdlambda(r) ight]^frac{1}{p}<infty.]
In Papers 5 we introduce and discuss a new class of linear operators on quasi-monotone sequences in $ell^2$. We give some characterizations for such a class, for instance we characterize the diagonal matrices.