Some new results concerning Lorentz sequence spaces and Schur multipliers : characterization of some new Banach spaces of infinite matrices

Abstract: This Licentiate thesis consists of an introduction and three papers, which deal with some new spaces of infinite matrices and Lorentz sequence spaces.In the introduction we give an overview of the area that serves as a frame for the rest of the thesis. In particular, a short description of Schur multipliers is given.In Paper 1 we prove that the space of all bounded operators on $\ell^2$ is contained in the space of all Schur multipliers on $B_w(\ell^2)$, where $B_w(\ell^2)$ is the space of linear (unbounded) operators on $\ell^2$ which map decreasing sequences from $\ell^2$ into sequences from $\ell^2$.In Paper 2 using a special kind of Schur multipliers and G. Bennett's factorization technique we characterize the upper triangular positive matrices from $B_w(\ell^p)$, $1In Paper 3 we consider the Lorentz spaces $\ell^{p,q}$ in the range $1\[\|x\|_{p,q}=\left(\sum_{n=1}^\infty (x^')^q n^{\frac{q}{p}-1}\right)^\frac{1}{q}\]is only a quasi-norm. In particular, we derive the optimal constant in the triangle inequality for this quasi-norm, which leads us to consider the following decomposition norm:\[\|x\|_{(p,q)}=\inf\{\sum_k \|x^{(k)}\|_{p,q}\},\]where the infimum is taken over all finite representations $x=\sum_k x^{(k)}$.