Banach spaces and operators which are nearly uniformly convex. (English) Zbl 0886.46017

Prus, Stanisław

Diss. Math. 363, 46 p. (1997).

Nearly uniform convexity introduced by R. Huff [Rocky Mountain J. Math. 10, 743-749 (1980; Zbl 0505.46011)] and nearly uniform smoothness are \(\sigma\)-local notions introduced for applications in the metric fixed point theory but very interesting also for its own. In this well written thesis, the author studies several aspects of these notions as well as their interaction with other metric properties of Banach spaces.
Chapter I contains the basic facts: A Banach space \(X\) is called to be nearly uniformly convex (NUC) if for every \(\varepsilon>0\) there is some \(\delta>0\) such that for each \(\varepsilon\)-distant sequence \((x_n)\) in \(B_X\) the inequality \(\inf\{|x|: x\in \text{co}\{x_n\}\}< 1-\delta\) holds true, and \(X\) is called nearly uniformly smooth (NUS) if for every \(\varepsilon>0\) there is some \(\eta>0\) such that for each \(0<\tau<\eta\) and for each basic sequence \((x_n)\) in \(B_X\) there is some \(k>1\) satisfying \(|x_1+\tau x_k|\leq 1+\varepsilon\tau\). Both notions are dual: \(X\) is NUC iff \(X^*\) is NUS.
In Chapter II a connection between NUS, NUC and \((p,q)\)-estimations is established. As a result of this kind it is shown that for all NUS-spaces \(X\) with shrinking \(M\)-basis \((v_n)\) there is some blocking \((X_k)\) of \((v_n)\) and some \(q>1\) such that \((X_k)\) satisfies \((\infty,q)\)-estimates. The NUC-spaces fulfil \((q,\infty)\)-estimates. Using this it can be shown that every \(X\) with NUC has an equivalent norm for which the modulus of convexity \(\Delta_X(\varepsilon)\) satisfies a power type estimate. Also a converse is true: If \(\lim_{\varepsilon\to 0}\Delta_X(\varepsilon)>0\) then \(X\) has an equivalent NUC-norm.
In Chapter III it is shown that the Lions-Peetre interpolation method preserves \((p,q)\)-estimates. This result is used in Chapter V to construct factorizations of operators through NUC-spaces.
In Chapter IV the notions of NUC- and NUS-operators are introduced and connections to weakly compact and Banach-Saks operators are established.

Reviewer: H.Junek (Potsdam)

MSC:

46B20	Geometry and structure of normed linear spaces
46B03	Isomorphic theory (including renorming) of Banach spaces
46B10	Duality and reflexivity in normed linear and Banach spaces
46B15	Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
47A05	General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
46B70	Interpolation between normed linear spaces

Keywords:

uniform convexity; nearly uniform smoothness; \(\sigma\)-local notions; metric fixed point theory; nearly uniformly convex; nearly uniformly smooth; \((p,q)\)-estimations; shrinking \(M\)-basis; Lions-Peetre interpolation method; factorizations of operators; weakly compact; Banach-Saks operators

Citations:

Zbl 0505.46011