Extreme 2-homogeneous polynomials on Hilbert spaces

Let H be a (real or complex) Hilbert space. We characterize the extreme points of the unit ball of the space of 2-homogeneous polynomials on H. We find the exact value of the λ-function for P(² H) and thus we show that its unit ball is the norm closed convex hull of its extreme points. We also describe topological properties of the set of extreme points, making connections between the set of extreme points and Grassmanian manifolds.     

Exposed 2-Homogeneous Polynomials on the two-Dimensional Real Predual of Lorentz Sequence Space

Mediterranean Journal of Mathematics - We classify the exposed polynomials of the unit ball of the space of 2-homogeneous polynomials on the two-dimensional real predual of Lorentz sequence space....     

A variational approach to norm attainment of some operators and polynomials

If X is an Asplund space, then every uniformly continuous function on Bx＊ which is holomorphic on the open unit ball, can be perturbed by a w＊ continuous and homogeneous polynomial on X＊ to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall＇s variational principle. We also show that the set of N-homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N-homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.     

Strongly Exposed Points in the Ball of the Bergman Space

We investigate which boundary points in the closed unit ball of the Bergman space A¹ are strongly exposed. This requires study of the Bergman projection and its kernel, the annihilator of Bergman space. We show that all polynomials in the boundary of the unit ball are strongly exposed.     

Geometry of homogeneous polynomials on two-dimensional real Hilbert spaces

We describe a method which characterises extreme points of the unit ball of the space of homogeneous polynomials of degrees 3 and 4 on two-dimensional Hilbert spaces and provides classes of examples of extreme and smooth polynomials of an arbitrary degree n.     

Multivariate Markov polynomial inequalities and Chebyshev nodes

This article considers the extension of V.A. Markov's theorem for polynomial derivatives to polynomials with unit bound on the closed unit ball of any real normed linear space. We show that this extension is equivalent to an inequality for certain directional derivatives of polynomials in two variables that have unit bound on the Chebyshev nodes. We obtain a sharpening of the Markov inequality for polynomials whose values at specific points have absolute value less than one. We also obtain an interpolation formula for polynomials in two variables where the interpolation points are Chebyshev nodes.     

Zeros of Polynomials on Banach Spaces: The Real Story

Let E be a real Banach space. We show that either E admits a positive definite 2-homogeneous polynomial or every 2-homogeneous polynomial on E has an infinite dimensional subspace on which it is identically zero. Under addition assumptions, we show that such subspaces are non-separable. We examine analogous results for nuclear and absolutely (1,2)-summing 2-homogeneous polynomials and give necessary and sufficient conditions on a compact set K so that C(K) admits a positive definite 2-homogeneous polynomial or a positive definite nuclear 2-homogeneous polynomial.     

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

Let H be a two-dimensional complex Hilbert space and P(³H){{\mathcal P}(^3H)} the space of 3-homogeneous polynomials on H. We give a characterization of the extreme points of its unit ball, B_P(³H){{\mathsf B}_{{\mathcal P}(^3H)}} , from which we deduce that the unit sphere of P(³H){{\mathcal P}(^3H)} is the disjoint union of the sets of its extreme and smooth points. We also show that an extreme point of B_P(³H){{\mathsf B}_{{\mathcal P}(^3H)}} remains extreme as considered as an element of B_L(³H){{\mathsf B}_{{\mathcal L}(^3H)}} . Finally we make a few remarks about the geometry of the unit ball of the predual of P(³H){{\mathcal P}(^3H)} and give a characterization of its smooth points.     

Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space

In an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.     

Unitarizable representations and fixed points of groups of biholomorphic transformations of operator balls

We show that the open unit ball of the space of operators from a finite-dimensional Hilbert space into a separable Hilbert space (we call it “operator ball”) has a restricted form of normal structure if we endow it with a hyperbolic metric (which is an analogue of the standard hyperbolic metric on the unit disc in the complex plane). We use this result to get a fixed point theorem for groups of biholomorphic automorphisms of the operator ball. The fixed point theorem is used to show that a bounded representation in a separable Hilbert space which has an invariant indefinite quadratic form with finitely many negative squares is unitarizable (equivalent to a unitary representation). We apply this result to find dual pairs of invariant subspaces in Pontryagin spaces. In Appendix A we present results of Itai Shafrir about hyperbolic metrics on the operator ball.     

Uniqueness of extensions of homogeneous polynomials on c0-sum of Hilbert spaces

We study the uniqueness of norm-preserving extension of n-homogeneous polynomials on X, where X is a c₀-sum of Hilbert spaces. We show that there exists a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on X to X″, but this result fails for homogeneous polynomials of degree greater than 2.     

Polynomial reflexivity in Banach spaces

We ask when the space ofN-homogeneous analytic polynomials on a Banach space is reflexive. This turns out to be related to whether polynomials are weakly sequentially continuous, and to the geometry of spreading models. For example, if these spaces are reflexive for allN, no quotient of the dual space may have a spreading model with an upperq-estimate, and every bounded holomorphic function on the unit ball has a Taylor series made up of weakly sequentially continuous polynomials (we assume the approximation property). Alencar, Aron and Dineen [AAD] gave the first example of some properties of a polynomially reflexive space (usingT*, the original Tsirelson space); we show that these properties and others are shared by all polynomially reflexive spaces. This paper forms a portion of the Ph. D. dissertation of the author, under the supervision of W. B. Johnson.     

Weakly continuous mappings on Banach spaces

It is shown that every n-homogeneous continuous polynomial on a Banach space E which is weakly continuous on the unit ball of E is weakly uniformly continuous on the unit ball of E. Applications of the result to spaces of polynomials and holomorphic mappings on E are given.     

Extreme operator-valued continuous maps

Let ?(H) denote the space of operators on a Hilbert spaceH. We show that the extreme points of the unit ball of the space of continuous functionsC(K, ?(H)) (K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dimH<∞ and cardK<∞, (b) exposed if and only ifH is separable andK carries a strictly positive measure.     

Smooth 2-homogeneous polynomials on Hilbert spaces

We determine the smooth points of the unit ball of the space of 2-homogeneous polynomials on a Hilbert space H. Working separately for the real and the complex cases we show that a smooth polynomial attains its norm. We deduce that the polynomial P is smooth if and only if there exists a unit vector x₀ in H such that P(x)=± á x,x₀ ñ ²+P₁(x₁)P(x)=\pm \left \langle x,x_{0}\right \rangle ^{2}+P_{1}(x_{1}) where x= á x,x₀ ñ x₀+x₁ x=\left \langle x,x_{0}\right \rangle x_{0}+x_{1} is the decomposition of x in H=span{ x₀} ?H₁H={\rm {span}}\{ x_{0}\} \oplus H_{1} and P₁ is a 2-homogeneous polynomial on H₁ of norm strictly less than 1.     

Local automorphisms of the Hilbert ball

Every holomorphic mapping which takes a piece of the boundary of the unit ball in complex Hilbert space into the boundary of the unit ball and whose differential at some point of this boundary is onto is the restriction of an automorphism of the ball. We also show that it is enough to assume that the mapping is only Gâteaux-holomorphic.

     

On the polynomial Lindenstrauss theorem

Under certain hypotheses on the Banach space X, we show that the set of N-homogeneous polynomials from X to any dual space, whose Aron–Berner extensions are norm attaining, is dense in the space of all continuous N-homogeneous polynomials. To this end we prove an integral formula for the duality between tensor products and polynomials. We also exhibit examples of Lorentz sequence spaces for which there is no polynomial Bishop–Phelps theorem, but our results apply. Finally we address quantitative versions, in the sense of Bollobás, of these results.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

Extreme operator-valued continuous maps

Let ℒ(H) denote the space of operators on a Hilbert spaceH. We show that the extreme points of the unit ball of the space of continuous functionsC(K, ℒ(H)) (K-compact Hausdorff) are precisely the functions with extremal values. We show also that these extreme points are (a) strongly exposed if and only if dimH<∞ and cardK<∞, (b) exposed if and only ifH is separable andK carries a strictly positive measure.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.