Numerical Index
by
Catherine Finet
Institut de Mathématique et d'Informatique, Université de Mons-Hainault, Belgium
Coauthors: M. Martin, R. Payá

The numerical index of a Banach space is a constant relating the norm and the numerical range of operators on the space. The numerical range of a bounded linear operator T on a Hilbert space H is the set W(T) = {(Tx|x), x in H, ||x|| = 1}. The concept of numerical range for operators on general Banach spaces was introduced independently by G. Lumer [L] and F. Bauer [B]. It is defined in the following way : the numerical range of a bounded linear operator T on a Banach space X is the set

V(T) = {x * (Tx), x in X, x * in X * , ||x|| = ||x * || = x * (x) = 1},

where X ^* is the dual space of X.

The numerical radius of T is given by

v(T) = sup { |\lambda|, \lambda in V(T)}.

And the numerical index of the space X is the constant :

n(X) = inf {v(T), ||T|| = 1}.

A survey on these notions and their relations to spectral theory of operators can be found in the books by F. Bonsall and J. Duncan [BD]. These notions have been intensively studied (see for example [LMP], [T]. Our aim is to study isomorphic properties of real Banach spaces with numerical index 1. For example, this happens with infinite dimensional reflexive real Banach spaces. Our aim is to show that, in this context, 1 is a very particular value for the numerical index.

This is a joint work with M. Martin, R. Payá [FMP].

References

[B] Bauer, F.L., On the field of values subordinate to a norm, Numer. Math., 4 (1962), 103-111.

[BF] Bonsall, F.F., Duncan, J., Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras", London Math. Soc. Lecture Note Series 2, Cambridge 1973.

[FMP] Finet, C., Martin, M., Payá, R., Numerical index and renorming. Preprint.

[L] Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc., 100 (1961), 29-43.

[LMP] López, G., Martin, M., Payá, R., Real Banach spaces with numerical ranges 1, Bull. London Math. Soc. 31 (1999), 207-212.

[T] Tillekeratne, K., Spatial numerical range of an operator, Proc. Camb. Phil. Soc. 76, (1974), 515-520.

Date received: March 6, 2001