Symmetric structures in banach spaces

Abstract

The backbone of the dissertation is a series of results to do with finding almost symmetric block bases of sequences which satisfy certain commonly occurring conditions. This has been known to be possible for some years: in 1982 and 1985 Amir and Milman published two important papers containing various results on the subject. My contribution has been to find substantially larger block bases and to construct examples in many cases to show that one cannot improve my new results further. Probabilistic methods play an important part in calculating bounds in both directions. These results were originally motivated by a theorem of Krivine, which can be regarded as the finite-dimensional analogue of the well-known distortion problem. Also in the dissertation is an analogue of the distortion problem in c₀, which strengthens considerably a result of James, and indicates that, contrary to what is generally believed, the answer to the distortion problem itself could very well be positive. There is also a counterexample to a fairly long-standing question about norm-attaining operators. I show that <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM SRC="/maths/ell.gif">p does not have property B if 1< p< <IMG WIDTH=14 HEIGHT=7 ALIGN=BOTTOM SRC="/maths/infinity.gif">. That is, I give an operator into <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM SRC="/maths/ell.gif">p which cannot be approximated in norm by a norm-attaining operator. In the last chapter, I give an unusual method of constructing the <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM SRC="/maths/ell.gif">p-spaces. By a very natural geometric process, one can build up a symmetric polytope that approximates the sphere to within sqrt2. The result generalizes to give polytopes that approximate the unit balls of the other <IMG WIDTH=8 HEIGHT=12 ALIGN=BOTTOM SRC="/maths/ell.gif">p-spaces. Given any function <IMG WIDTH=7 HEIGHT=12 ALIGN=BOTTOM SRC="/maths/lambda.gif">:N<IMG WIDTH=14 HEIGHT=9 ALIGN=BOTTOM SRC="/maths/rightarr.gif">R that satisfies <IMG WIDTH=7 HEIGHT=12 ALIGN=BOTTOM SRC="/maths/lambda.gif">(k) = <IMG WIDTH=5 HEIGHT=23 ALIGN=BOTTOM SRC="/maths/capvert.gif"><IMG WIDTH=9 HEIGHT=11 ALIGN=BOTTOM SRC="/maths/capsigma.gif">k_{1 xi <IMG WIDTH=5 HEIGHT=23 ALIGN=BOTTOM SRC="/maths/capvert.gif"> for some symmetric basis x_1, x_2,..., one can use the same process to give a natural example of such a basis. Some elementary properties of this class of spaces are investigated.}