RANDOM BANACH-SPACES - THE LIMITATIONS OF THE METHOD

We shall study the properties of typical n-dimensional subspaces of l( infinity)(N) = (R(N), parallel to.parallel to(infinity)), or equivalen tly, typical n-dimensional quotients of l(1)(N) = (R(N), parallel to.p arallel to(<l>)), where the meaning of what is typical and what is not is defined in terms of the Haar measure mu(n,N) on the Grassmann mani fold G(n,N) of all n-dimensional subspaces of R(N). In [G1.2), Gluskin proved that a ''typical'' n-dimensional subspace E of l(infinity)(n2) enjoys the property parallel to P parallel to greater than or equal t o ck/root n log n, for every projection P: E-->E, with min {rank P, ra nk (Id - P)} = k, where c is a numerical constant. In particular, if k greater than or equal to n(alpha), alpha > 1/2, then no projection P on E with both rank P and corank P greater than k can be ''well'' boun ded. Several other results, [Sz.1], [Sz.2], [Ma.1], [Ma.2] showed that a ''typical'' proportional (i.e., dim E approximate to beta N for som e ''fixed'' beta is an element of(0, 1)) subspace E of l(infinity)(N) has the property that every ''well'' bounded operator on E is indeed a ''small'' perturbation of a multiple of the identity lambda Id(E). Ho wever, the estimates on the distance between T and lambda Id(E) have b een made in terms of the geometry of R(N) rather than E itself. In thi s note, we obtain the estimates on the distance between T and lambda I d(E) in intrinsic terms of the geometry of E, namely, in terms of the Gelfand numbers of T-lambda Id(E) (Sections 2 and 3). On the other han d, we show in Section 4, that if k less than or equal to n(1/2) then a ''typical'' n-dimensional subspace E of l(infinity)(N) (for any N gre ater than or equal to n) contains a k-dimensional well-complemented su bspace G isomorphic to l(p)(k) with either p = 2 or p = infinity and t herefore admits operators which are ''fairly'' far away from the line (lambda Id(E))lambda epsilon R. We shall employ the standard notation of local theory of Banach spaces as used in e.g., [F-L-M]. For basics on multivariate Gaussian random variables the reader is referred to [T ].