| Research Abstract |
Geometrical properties of Banach spaces as well as related norm inequalities are investigated from an operator theoretical point of view. This approach allows a unifying treatment of them and also enables us to apply interpolation techniques in research of the Banach space geometry. Not only they have their own beauty and significance, but also they provide essential or useful notions and tools in various branches of analysis including applicable areas, which indicates the fundamental importance of this subject.
Major results are as follows.
1. On Clarkson-type inequalities :
(1) A sequence of Clarkson-type inequalities are characterized in the general Banach space setting by the notions of Rademacher type and cotype which are of great importance in "Probability in Banach Spaces".
(2) It is shown how Clarkson's and related inequalities are inherited by the Lebesgue-Bochner space L_r (X) from a given Banach space X, by which most of these inequalities known for various spaces are derived unifyingly.
2. On the von Neumann-Jordan (NJ-) constant of a Banach space a sort of modulus of skewness of the norm :
(1) A systematic way to calculate NJ-constant is given, by which all the previous results for various spaces and some new ones as well are obtained.
(2) A sequence of informations NJ-constant gives is presented, especially about type and cotype, uniform convexity, uniform non-squareness, super-reflexivity, normal structure and fixed point property, etc.
3. Several geometrical properties are charcterized unifyingly via behavior of operator norms of 1 matrices between finite dimensional X-valued l_p-spaces. In particular, a sequence of characterizations of uniformly non-square spaces is given, some of which are similar to the well-known one for uniform convexity.
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