CONTENTSIntroduction..............................................................................................5 I. Basic definitions and notation..............................................................6 M-bases and finite-dimensional decompositions....................................6 Some geometric properties of Banach spaces.......................................9 II. Constructions of equivalent norms.....................................................12 III. (p,q)-estimates in interpolation...
CONTENTSIntroduction................................................................................................................................5I. Basic notations and definitions................................................................................................7II. Basic properties of finite-dimensional decompositions with (p,q)-estimates............................8III. A construction of f.d.d.'s satisfying (p,q)-estimates and its geometric applications...............13IV. An...
An infinite dimensional counterpart of uniform smoothness is studied. It does not imply reflexivity, but we prove that it gives some -type estimates for finite dimensional decompositions, weak Banach-Saks property and the weak fixed point property.
It is shown that if a Banach space X has the weak Banach-Saks property and the weak fixed point property for nonexpansive mappings and Y has the asymptotic (P) property (which is weaker than the condition WCS(Y) > 1), then X ⊕ Y endowed with a strictly monotone norm enjoys the weak fixed point property. The same conclusion is valid if X admits a 1-unconditional basis.
Logarithmic convexity of a measure of weak noncompactness for bounded linear operators under Calderón’s complex interpolation is proved. This is a quantitative version for weakly noncompact operators of the following: if T: A₀ → B₀ or T: A₁ → B₁ is weakly compact, then so is for all 0 < θ < 1, where and are interpolation spaces with respect to the pairs (A₀,A₁) and (B₀,B₁). Some formulae for this measure and relations to other quantities measuring weak noncompactness are established.
We give an example of a Banach lattice with a non-convex modulus of monotonicity, which disproves a claim made in the literature. Results on preservation of the non-strict Opial property and Opial property under passing to general direct sums of Banach spaces are established.
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