Polynomials on Banach Spaces with Unconditional Schauder Bases

Let X be a Banach space with an unconditional Schauder basis. We consider the problem of representing a homogeneous polynomial on a X by an unconditionally convergent monomial expansion. A recent result of Defant and Kalton shows that, in infinite dimensions, the space of n-homogeneous polynomials does not have an unconditional basis. However, it may be possible to represent a homogeneous polynomial by a monomial expansion that is pointwise unconditionally convergent. For example, when X = ℓ_{1}, every homogeneous polynomial has this property. Matos isolated the spaces of n-homogeneous polynomials that have pointwise unconditionally convergent monomial expansions and defined a complete norm for these spaces. We explore some properties of the Matos spaces, including a connection with absolutely summing operators.

Ray Ryan, National University of Ireland, Galway.