We exhibit abelian topological groups admitting no nontrivial strongly continuous irreducible representations in Banach spaces. Among them are some abelian Banach–Lie groups and some monothetic subgroups of the unitary group of a separable Hilbert space.

Let be a proper l.s.c. convex function on a real Hilbert space H. We show that if H is separable, then 4> is twice differentiate in some sense on a dense subset of the graph of d.

In this note I am writing out some of the material from Paul Halmos, Hilbert Space Problem Book, on shift operators. The reason I’m doing this is because shift operators are standard objects in operator theory and every analyst should know their properties and their spectra. A reference to Problem n of Halmos is a reference to Problem n in this book. An orthonormal basis for a Hilbert space is ...

A geometrical picture of separability of 2× 2 composite quantum systems, showing the region of separable density matrices in the space of hermitian matrices, is given. It rests on the criterion of separability given by Peres [1], and it is an extension of the “Horodecki diagram” [2] and the “stella octangula” described by Aravind [3]. An illustration of which composite quantum states that are s...

Let $\mathcal H $ be a complex, separable Hilbert space, and B(\mathcal H) denote the set of all bounded linear operators on $. Given an orthogonal projection $P \in \mathcal operator $D B(\mathc

In this paper, we study spaces of vector functions that are holomorphic in the angular domain complex plane and with values a separable Hilbert space. We show that, equipped appropriate norms, these spaces.

M. Gromov [7] suggested to use coarse embeddings into a Hilbert space or into a uniformly convex space as a tool for solving some of the well-known problems. G. Yu [21] and G. Kasparov and G. Yu [11] have shown that this is indeed a very powerful tool. On the other hand, there exist separable metric spaces ([6] and [5, Section 6]) which are not coarsely embeddable into Hilbert spaces. In [9] (s...

In this paper, we study regression problems over a separable Hilbert space with the square loss, covering non-parametric regression over a reproducing kernel Hilbert space. We investigate a class of spectral-regularized algorithms, including ridge regression, principal component analysis, and gradient methods. We prove optimal, high-probability convergence results in terms of variants of norms ...

The paper considers continuity properties of a nite Borel measure on a separable Hilbert space X. Continuity of the mappings x 7 ! (A + x) on a subspace H, uniformly in A 2 B(X), is characterized by the existence of a nonstandard density of having a certain property. This generalizes a well known standard result for measures on R n. The ideas are illustrated with reference to Gaussian measures ...

Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space Rn, Hilbert spaces admit various useful realizations as spaces of functions. In the paper this simple observation is used to construct a fruitful formalism of local coordinates on Hilbert manifolds. ...