Powers of Symmetric Sequence Ideals

About 50 years ago it was observed that the relationship between L-1 and L-p can be mimicked by assigning to certain function spaces A a "power" A(p). We study this construction in the case of symmetric sequence ideals a by letting a(p) := {s is an element of l(infinity) : vertical bar s vertical bar(1/p) is an element of a} for all p > 0. The purely algebraic level is simple. However, if a is equipped with a norm parallel to . vertical bar a parallel to, then parallel to s vertical bar a(p)parallel to := parallel to vertical bar s vertical bar(1/p)vertical bar a parallel to(p) for all s is an element of a(p) need not be a norm. This defect disappears in the setting of quasi-norms. Therefore quasi-Banach power scales {a(p)}(p>0) are well-behaved subjects, which deserve to be studied. In this paper, we are mainly dealing with {a(p)}(p>0) as a whole and not so much with the properties of its single members a(p). Our main purpose is to improve the knowledge about the structure of the ( very large) lattices of all symmetric sequence ideals. So specific one-parametric subsets, which look like intervals, are important. This paper should be a good illustration of the leitmotiv: Well-chosen concepts make mathematics easy.