Geometry of 2-homogeneous polynomials on Hilbert spaces
by
Bogdan C. Grecu
National University of Ireland, Galway

Let H be a Hilbert space. We determine the extreme and the smooth points of the unit ball of the space of 2-homogeneous polynomials on H. In dealing with the extreme points, on a real Hilbert space we show that the polynomial P is extreme if and only if there exists an orthogonal decomposition H=H₁\oplusH₂ with associated orthogonal projections \pi₁ and \pi₂ such that

P(x)=|| \pi₁x|| ²-|| \pi₂x|| ².

In the complex case we prove that P is extreme if and only if there exists an orthonormal basis {e_j}_{j in J} for H such that P(x)=\sum_{j in J}x_j². In both cases the spectral theorem for self adjoint operators plays a central part. When the (complex or real) space H is infinite dimensional we also show that the space of 2-homogeneous approximable polynomials is not a dual space.

We then determine the smooth points. Working separately for the real and the complex case we show that a smooth polynomial attains its norm. When H is complex we use again the spectral theorem to show that a 2-homogeneous polynomial factors through a complexification of a real 2-homogeneous polynomial on the underlying real structure of H. Then we deduce that the polynomial P is smooth if and only if there exists a unit vector x₀ in H such that H=span{x₀}\oplusH₁ and P(x)= +/- á x, x₀ ñ ²+P₁(x₁) where x= á x, x₀ ñ x₀+x₁ is the decomposition of x and P₁ is a 2-homogeneous polynomial on H₁ of norm strictly less than 1.

(T)

Date received: November 29, 1999