OPERATOR SUPERPOSISI DAN OPERATOR URUTAN PADA RUANG BARISAN BERNILAI RIESZ; SUPERPOSITION OPERATORS AND ORDERED OPERATORS ON RIESZ VALUED SEQUENCE SPACES

Herawati, Elvina, Supama

2015 | Tesis | FMIPA UGM

In this dissertation we introduce a new concept about Riesz-valued sequence spaces thats defined by ordered modular by using an operator in Riesz spaces called order-? function. This spaces are generalization of vector valued Orlicz sequence spaces. We begin by introducing some fundamental concepts relate to Riesz spaces, such as some properties of elements in Riesz spaces, order convergence and order Cauchy sequences in Riesz spaces and their characterization, normed Riesz spaces, and Banach lattices. Let E be a Riesz space. A space of E-valued sequences will be denoted by X(E). Furthermore, we introduce an order ?-function ? and define the space X9(E; ?). If the function ? is convex and X(E) is an ideal normed Riesz space, we get the space X9(E; ?) is an ideal normed Riesz space with respect to norm k?k?, that is a normed defined by ordered mudular. Some monotonic properties of the norm k?k?, such as strictly monotonicity, uniformly monotonicity and locally uniformly monotonicity, are observed as well. We also investigate some topological properties of the space X9(E; ?). Furthermore, we define a superposition operator Pg on X9(E; ?) using a generating function g : N ? E ! E, where E is a normed Riesz space ?-Dedekind complete. Based on the properties of the function g, we formulate necessary and sufficient conditions so that ther superposition operator Pg maps the space X9(E; ?) into the Riesz valued sequence space `(E; ?), especially for X is a classic real valued sequence spaces c0 or `1, where c0 and `1 are the space real valued sequences that converge to zero and absolutely summable respectively. Futhermore, we prove that the continuity of g is a necessary and sufficient condition for the continutity of the superposition operator Pg. In this dissertation we also discuss an ordered operator T : E ! F, for E and F are Riesz spaces with F is a Dedekind complete-? with stable properties. We also formulate the necessary and sufficient conditions so that the ordered operator maps the special Riesz valued sequence spaces, i.e., from c(E) to c(F), from `1(E) to `1(F) and from `1(E) to c(F). We also derive necessary conditions so that the ordered operator maps `1(E) into c(E). In the last part of this dissertation, we formulate necessary and sufficient conditions so that the ordered operator T maps X(E) into c(E), where X(E) is a BLK space satisfies the AK property. In case X(E) is a BLK space, we can derive necessary and sufficient conditions so that the ordered operator T maps X(E) into `1(E) and T maps X(E) into `1(E). Keywords : Ordered convergent,

Kata Kunci : Ordered convergent; Banach lattice; strictly monotone; uniformly monotone; uniformly local monotone; superposition operator; ordered operator; BLK space; AK properties.