The unit ball of the complex $${{\mathcal P}(^3H)}$$

Abstract

Let H be a two-dimensional complex Hilbert space and P( 3 H ) the space of 3-homogeneous polynomials on H . We give a characterization of the extreme points of its unit ball, B P( 3 H ) , from which we deduce that the unit sphere of P( 3 H ) is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B P( 3 H ) remains extreme as considered as an element of B L( 3 H ) . Finally we make a few remarks about the geometry of the unit ball of the predual of P( 3 H ) and give a characterization of its smooth points. Notation, terminology and preliminary resultsGiven n (real or complex) Banach spaces X 1 , . . . , X n we denote by X 1 ⊗· · ·⊗ X n their tensor product and by π the projective norm. If X 1⊗π · · ·⊗ π X n is the completion of X 1 ⊗· · · ⊗ X n under the projective norm, then we have (X 1⊗π · · ·⊗ π X n ) * = L( n X 1 , . . . , X n ), the space of continuous n-linear forms on X 1 × · · · × X n endowed with the supremum norm. However, as remarked in [6], most of the times the multilinear theory is far from being just a simple

Geometry of Banach spaces of trinomials

“…In a similar direction, Choi and Kim [2][3][4] considered the same problem for scalar-valued 2-homogeneous polynomials on the real spaces 2 1 , 2 2 and 2 ∞ whereas Grecu [7] treated the same question for scalarvalued 2-homogeneous polynomials on the real spaces 2 p with 1 < p < ∞. See also [5][6][7][8][9][10] for related questions concerning real or complex homogeneous polynomials of degree 2 or 3. Trigonometric trinomials (real or complex) have also been studied by Aron and Klimek [1], Neuwirth [14] and Révész [15].…”

Smooth 2-homogeneous polynomials on the plane with a hexagonal norm

“…Thus we fully described the geometry of the unit ball of P( 2 d * (1, w) 2 ). We refer to [1,8,18,19] and references therein for some recent work about extremal properties of homogeneous polynomials on some classical Banach spaces.…”

Extreme and exposed symmetric bilinear forms on the space ${\mathcal L}_{s}(^2 l_{\infty}^2)$

“…, respectively. We refer to [1,2,5,6,[9][10][11][13][14][15]20,23,26,27,31,36,[38][39][40][41][42][43][44][45][46][47] for some recent work about extremal properties of homogeneous polynomials and multilinear forms on Banach spaces.…”