Albrecht Pietsch wrote an excellent book on operator ideals in 1978, which I use frequently. But, I was reading the preface to his book, and I do not understand it. He writes (in English translation):
In this monograph we present the theory of operator ideals which, beginning with the fundamental work of A. Grothendieck, is becoming more and more a special branch of functional analysis producing results and problems of its own interest. On the other hand, we would like to give a lot of remarkable applications to the spectral theory of operators, to the geometry of Banach spaces, and to the theory of stochastic processes. Last, but not least, we shall obtain a considerably large part of the theory of nuclear locally convex spaces within this new context.
I don't see any connection given in his book to stochastic processes, nor to spectral theory. I'm also unsure of what he means by "the geometry of Banach spaces."
Let us say that $\mathcal{J}$ is an operator ideal--or, perhaps, a Banach ideal. What, specifically, can that tell us about the geometry of a given space $X$?
So, my question is four-fold:
(1) What is the connection between operator ideals and spectral theory?
(2) What--specifically--is the connection between operator ideals and the geometry of Banach spaces.
(3) What is the relationship between operator ideals and stochastic processes?
(4) What is the connection between operator ideals and the theory of nuclear locally convex spaces.
Thank you.