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Let H:=... denote the Heisenberg group with the usual Carnot-Caratheodory metric d. It is known (since the work of Pansu and Semmes) that the metric space (H, d) cannot be embedded in a bilipschitz fashion into a Hilbert space; however, from a general theorem of Assouad, for any 0 < ε ≤ 1/2, the snowflaked metric space (H, d1-ε) embeds into an infinite-dimensional Hilbert space with distortion O(e-1/2). This distortion bound was shown by Austin, Naor, and Tessera to be sharp for the Heisenberg group H. Assouad's argument allows I² to be replaced by RD(ε) for some dimension D(ε) dependent on ε. Naor and Neiman showed that D could be taken independent of ε, at the cost of worsening the bound on the distortion to O(ε-1-cD ), where cD → as D → ∞. In this paper we show that one can in fact retain the optimal distortion bound O(ε-1/2) and still embed into a bounded-dimensional space RD, answering a question of Naor and Neiman. As a corollary, the discrete ball of radius R ≥ 2 in r :=... can be embedded into a bounded-dimensional space RD with the optimal distortion bound of O(log1/2 R). The construction is iterative, and is inspired by the Nash-Moser iteration scheme as used in the isometric embedding problem; this scheme is needed in order to counteract a certain "loss of derivatives" problem in the iteration. [ABSTRACT FROM AUTHOR]

Abstract: A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T. In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions. [Copyright &y& Elsevier]

The purpose of this paper is to prove that if the Pták function p is an operator norm, on \mathcal{B}(E), associated to a norm | . |, then (E, | . |) is a pseudo-Hilbert space. As a consequence, we obtain that if \mathcal{B}(E) is a C*-algebra, then E is a Hilbert space.

In this paper, we propose the study of a conjecture whose positive solution would provide an example of a non-convex Chebyshev set in an infinite-dimensional real Hilbert space.