Quantitative versions of hereditary results on M-ideals of compact operators
by
Rainis Haller
University of Tartu
Coauthors: Eve Oja (University of Tartu), Eckart Plewnia

It is well-known that a Banach space X is an M-ideal in its bidual whenever the space K(X, X) of compact operators on X is an M-ideal in the space L(X, X) of bounded operators. The same conclusion holds whenever the space K(l₁, X) of compact operators from l₁ to X is an M-ideal in L(l₁, X).

We develop a unified approach for both of these hereditary results in the context of ideals satisfying the M(r, s)-inequality. We also study quantitative aspects of these results.

Let r, s in (0, 1]. A (closed) subspace X of a Banach space Y is called an ideal satisfying the M(r, s)-inequality in Y if there exists a norm-one projection P on the dual space of Y ^* with kerP=X ^\perp (the annihilator of X in Y ^* ) (i.e. X is an ideal in Y) and ||y ^* || >= r||Py ^* ||+s||y ^* -Py ^* || for every y ^* in Y ^* .

Clearly, X is an ideal satisfying the M(1, 1)-inequality in Y if and only if X is an M-ideal in Y.

Our main theorem shows the following. Let X, Y, and Z be Banach spaces, X being simultaneously isomorphic to a quotient space of Z and a subspace of Y. If K(Z, Y) satisfies the M(r, s)-inequality in L(Z, Y) for some projection P, then (under some restrictions involving uniqueness of norm-preserving extensions of some kind of functionals) X satisfies the M(r/j(X, Y), s/j(X, Y) q(Z, X))-inequality in its bidual, where j(X, Y)=inf{|| T||/j(T)  :  T in L(X, Y) isomorphism ``in''} and q(Z, X)=inf{|| Q||/q(Q)  :  Q in L(Z, X) surjection} with j(T) and q(Q) denoting the injection modulus of T and the surjection modulus of Q. For isomorphic X, Y, and Z, this immediately implies that X satisfies the M(r/d(X, Y), s/d(Z, X) d(X, Y))-inequality, where d(·, ·) denotes the Banach-Mazur distance.

(P)

Date received: March 31, 2000