We develop several methods of realization of scalar product and generalized moment problems. Constructions are made by use of a Hilbertian method or a fixed point method. The constructed solutions are rational fractions and exponentials of polynomials. They are connected to entropy maximization. We give the general form of the maximizing solution. We show how it is deduced from the maximizing solution of the algebraic moment problem.

Let T be a precompact subset of a Hilbert space. We estimate the metric entropy of co(T), the convex hull of T, by quantities originating in the theory of majorizing measures. In a similar way, estimates of the Gelfand width are provided. As an application we get upper bounds for the entropy of co(T), $T=t_1,t_2,...$, $\left|\right|t_j\left|\right|\le a_j$, by functions of the $a_j$’s only. This partially answers a question raised by K. Ball and A. Pajor (cf. [1]). Our estimates turn out to be optimal in the case of slowly decreasing sequences ${\left(a_j\right)}_{j=1}^{\infty }$.

Let ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ be the set of partial isometries with finite rank of an infinite dimensional Hilbert space $\mathcal{H}$. We show that ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a smooth submanifold of the Hilbert space ${\mathcal{B}}_2\left(\mathcal{H}\right)$ of Hilbert-Schmidt operators of $\mathcal{H}$ and that each connected component is the set ${\mathcal{I}}_N$, which consists of all partial isometries of rank $N<\mathrm{\infty }$. Furthermore, ${\mathcal{I}}_{\left(\mathrm{\infty }\right)}$ is a homogeneous space of ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}\times {\mathcal{U}}_{\left(\mathrm{\infty }\right)}$, where ${\mathcal{U}}_{\left(\mathrm{\infty }\right)}$ is the classical Banach-Lie group of unitary operators of $\mathcal{H}$, which are Hilbert-Schmidt perturbations of the identity. We introduce two Riemannian metrics...

In this paper we characterize Moore-Penrose inverses of Gram matrices leaving a cone invariant in an indefinite inner product space using the indefinite matrix multiplication. This characterization includes the acuteness (or obtuseness) of certain closed convex cones.

We use the Maurey-Rosenthal factorization theorem to obtain a new characterization of multiple 2-summing operators on a product of $l_p$ spaces. This characterization is used to show that multiple s-summing operators on a product of $l_p$ spaces with values in a Hilbert space are characterized by the boundedness of a natural multilinear functional (1 ≤ s ≤ 2). We use these results to show that there exist many natural multiple s-summing operators $T:l_{4/3}\times l_{4/3}\to l₂$ such that none of the associated linear operators is s-summing...

For a Banach space X, we show how the existence of a norm-one element u in X and a norm-one continuous bilinear mapping f: X x X --&gt; X satisfying f(x,u) = f(u,x) = x for all x in X, together with some more intrinsic conditions, can be utilized to characterize X as a member of some relevant subclass of the class of Banach spaces.