# Some new results concerning Banach spaces of infinite matrices and Lorentz sequence spaces

Abstract: This PhD thesis consists of an introduction and five 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 sequences decreasing from $ ell ^ $ 2 into sequences from $ ell ^ 2 $. In Paper 2 using a special kind of Schur multipliers and G. Bennett 's actually Riza tion technique we characterize the upper triangular matrices positive from $ B_w ( ell ^ p) $, $ 1 In Paper 3 we consider the Lorentz spaces $ ell ^ (p, q) $ in the range $ 1 [ | x | _ (p, q) = left ( sum_ (n = 1) ^ infty (x_n ^ ') ^ qn ^ ( 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 rule: [ | x | _ ((p, q)) = inf ( sum_k | x ^ ((k)) | _ (p, q) ); ] Where the Infimum is taken over all finite representation $ x = sum_k x ^ ((k)) $. In Paper 4 we denotes by $ B_p ( ell ^ 2) $ the Besov-Schatten space of all upper triangular matrices $ A $ such that [ | A | _ (B_p ( ell ^ 2)) = left [ int_0 ^ 1 (1-r ^ 2) ^ (2p) | A''(r) | _ (C_p) ^ pd lambda (r) right] ^ frac (1) (p) < infty. ] and we prove a natural relationship between the Bergman projection and the Besov-Schatten spaces. In Paper 5, given a matrix $ A $ satisfying that [ Ax in ell ^ p text (for every) x in ell ^ p text (with) | x_k | searrow 0 ] For $ 1 leq p < infty $, we show that [ A in B_w ( ell ^ p) text (if and only if) sup_ (n in mathbb N ^ ') left ( frac (1) (n) sum_ (k = 1) ^ n | a_k | ^ p right) ^ frac (1) (p) < infty ] For $ A = a_0 $ given by $ a = (a_k) _ (k in mathbb (N)) $. We prove that there exist some classes of operators either Belonging to $ B_w ( ell ^ p) $ or to the space of all Schur multipliers on $ B_w ( ell ^ p) $.

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