Atlas: Quantitative versions of hereditary results on $M$-ideals of compact operators by Rainis Haller

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Functional Analysis Valencia 2000
July 3-7, 2000
Technical University of Valencia (UPV) and University of Valencia (UV)
Valencia, Spain

Organizers
R.M. Aron (Kent State U., USA), K.D. Bierstedt (U. Paderborn, Germany), J. Bonet (UPV), J. Cerdà (U. Barcelona, Spain), H. Jarchow (U. Zürich, Switzerland), M. Maestre (UV), J. Schmets (U. Liège, Belgium)

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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