Search results for "BERT"

Bessel sequences, Riesz-like bases and operators in Triplets of Hilbert spaces

2016

Riesz-like bases for a triplet of Hilbert spaces are investigated, in connection with an analogous study for more general rigged Hilbert spaces performed in a previous paper. It is shown, in particular, that every \(\omega \)-independent, complete (total) Bessel sequence is a (strict) Riesz-like basis in a convenient triplet of Hilbert spaces. An application to non self-adjoint Schrodinger-type operators is considered. Moreover, some of the simplest operators we can define by them and their dual bases are studied.

Some representation theorems for sesquilinear forms

2016

The possibility of getting a Radon-Nikodym type theorem and a Lebesgue-like decomposition for a non necessarily positive sesquilinear $\Omega$ form defined on a vector space $\mathcal D$, with respect to a given positive form $\Theta$ defined on $\D$, is explored. The main result consists in showing that a sesquilinear form $\Omega$ is $\Theta$-regular, in the sense that it has a Radon-Nikodym type representation, if and only if it satisfies a sort Cauchy-Schwarz inequality whose right hand side is implemented by a positive sesquilinear form which is $\Theta$-absolutely continuous. In the particular case where $\Theta$ is an inner product in $\mathcal D$, this class of sesquilinear form cov…

A description of pseudo-bosons in terms of nilpotent Lie algebras

2017

We show how the one-mode pseudo-bosonic ladder operators provide concrete examples of nilpotent Lie algebras of dimension five. It is the first time that an algebraic-geometric structure of this kind is observed in the context of pseudo-bosonic operators. Indeed we don't find the well known Heisenberg algebras, which are involved in several quantum dynamical systems, but different Lie algebras which may be decomposed in the sum of two abelian Lie algebras in a prescribed way. We introduce the notion of semidirect sum (of Lie algebras) for this scope and find that it describes very well the behaviour of pseudo-bosonic operators in many quantum models.

On distinguished polynomials and their projections

2012

We study projections and injections between projective tensor products spaces or spaces of polynomials and we show that the example of a polynomial constructed in (4), that is neither p-dominated nor compact, can be identified with the projection map of the symmetric tensor product onto the space. Also we give a characterization of the weak and quasi approximation properties on symmetric tensor products.

A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting

2015

Abstract A Kurzweil-Henstock type integral with respect to an abstract derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using technical properties of Riesz spaces.

PIP-Space Valued Reproducing Pairs of Measurable Functions

2019

We analyze the notion of reproducing pairs of weakly measurable functions, a generalization of continuous frames. The aim is to represent elements of an abstract space Y as superpositions of weakly measurable functions belonging to a space Z : = Z ( X , μ ), where ( X , μ ) is a measure space. Three cases are envisaged, with increasing generality: (i) Y and Z are both Hilbert spaces; (ii) Y is a Hilbert space, but Z is a pip-space; (iii) Y and Z are both pip-spaces. It is shown, in particular, that the requirement that a pair of measurable functions be reproducing strongly constrains the structure of the initial space Y. Examples are presented for each case.

Normed Quasi *-Algebras: Bounded Elements and Spectrum

2020

Bounded elements of a Banach quasi *-algebra are intended to be those, whose images under every *-representation are bounded operators in a Hilbert space. This rough idea can be developed in several ways, as we shall see in the present chapter. These notions lead us to discuss a convenient concept of spectrum of an element in this context.

THE BISHOP-PHELPS-BOLLOBAS PROPERTY FOR HERMITIAN FORMS ON HILBERT SPACES

2013

On non-self-adjoint operators defined by Riesz bases in Hilbert and rigged Hilbert spaces

2018

In this paper we discuss some results on non self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that their eigenvectors form Riesz bases of a certain Hilbert space. Also, we exhibit a generalization of those results to the case of rigged Hilbert spaces, and we also consider the problem of the factorization of the aforementioned Hamiltonians in terms of generalized lowering and raising operators.

TALPID3 controls centrosome and cell polarity and the human ortholog KIAA0586 is mutated in Joubert syndrome (JBTS23)

2015

Joubert syndrome (JBTS) is a severe recessive neurodevelopmental ciliopathy which can affect several organ systems. Mutations in known JBTS genes account for approximately half of the cases. By homozygosity mapping and whole-exome sequencing, we identified a novel locus, JBTS23, with a homozygous splice site mutation in KIAA0586 (alias TALPID3), a known lethal ciliopathy locus in model organisms. Truncating KIAA0586 mutations were identified in two additional patients with JBTS. One mutation, c.428delG (p.Arg143Lysfs*4), is unexpectedly common in the general population and may be a major contributor to JBTS. We demonstrate KIAA0586 protein localization at the basal body in human and mouse p…