Unconditional Basis and Gordon–Lewis Constants for Spaces of Polynomials
✍ Scribed by Andreas Defant; Juan Carlos Dı́az; Domingo Garcı́a; Manuel Maestre
- Publisher
- Elsevier Science
- Year
- 2001
- Tongue
- English
- Weight
- 228 KB
- Volume
- 181
- Category
- Article
- ISSN
- 0022-1236
No coin nor oath required. For personal study only.
✦ Synopsis
No infinite dimensional Banach space X is known which has the property that for m 2 the Banach space of all continuous m-homogeneous polynomials on X has an unconditional basis. Following a program originally initiated by Gordon and Lewis we study unconditionality in spaces of m-homogeneous polynomials and symmetric tensor products of order m in Banach spaces. We show that for each Banach space X which has a dual with an unconditional basis (x i *), the approximable (nuclear) m-homogeneous polynomials on X have an unconditional basis if and only if the monomial basis with respect to (x i *) is unconditional. Moreover, we determine an asymptotically correct estimate for the unconditional basis constant of all m-homogeneous polynomials on l n p and use this result to narrow down considerably the list of natural candidates X with the above property.