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Giorgia Bellomonte

Publications and source records attributed to Giorgia Bellomonte.

14 recordsLinked to original sources
arXiv2026

Cauchy-Schwarz inequalities for maps in noncommutative Lp-spaces

In this paper, a generalized Cauchy-Schwarz inequality for positive sesquilinear maps with values in noncommutative Lp-spaces for p > 1 are obtained. Bound estimates for their real and imaginary parts are also provided, and, as an application, a generalization of the uncertainty relation in the context of noncommutative L2-spaces are given. Next, a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von Neumann algebra into a C*-algebra equipped with the numerical radius norm is proved. In the same spirit, a new norm on a noncommutative L2-space, which generalizes the classical numerical radius norm of bounded linear operators on a Hilbert space, is proposed, and a Cauchy-Schwarz inequality for positive sesquilinear maps with values in the space of bounded linear operators from a von-Neumann algebra into the noncommutative L2-space equipped with this new norm is proved. These results are used to get representations of general positive linear maps with values into a non-commutative Lp-space and into certain operator spaces in several different situations. Some concrete examples are also given.

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arXiv2025

Banach bimodule-valued positive maps: Inequalities and induced representations

In this paper, we consider representations induced by general positive and completely positive sesquilinear maps with values in ordered Banach bimodules, such as the space of trace-class operators and the spaces of bounded linear operators from a von Neumann algebra into the dual of another von Neumann algebra. Also, we deduce some new inequalities for these maps.

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arXiv2024

On representations and topological aspects of positive maps on non-unital quasi *-algebras

In this paper, we provide a representation of a certain class of C*-valued positive sesquilinear and linear maps on non-unital quasi *-algebras. Also, we illustrate our results on the concrete examples of non-unital Banach quasi *-algebras, such as the standard Hilbert module over a commutative C*-algebra, Schatten p-ideals, and noncommutative L2 spaces induced by a semifinite, nonfinite trace. As a consequence of our results, we obtain a representation of all bounded positive linear C*-valued maps on non-unital C*-algebras. We also deduce some norm inequalities for these maps. Finally, we consider a noncommutative L2 space equipped with the topology generated by a positive sesquilinear form and we construct a topologically transitive operator on this space.

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arXiv2023

Topological aspects of quasi *-algebras with sufficiently many *-representations

Quasi *-algebras possessing a sufficient family $\mathcal{M}$ of invariant positive sesquilinear forms carry several topologies related to $\mathcal{M}$ which make every *-representation continuous. This leads to define the class of locally convex quasi GA*-algebras whose main feature consists in the fact that the family of their bounded elements, with respect to the family $\mathcal{M}$, is a dense C*-algebra.

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arXiv2022

Frame-related Sequences in Chains and Scales of Hilbert Spaces

Frames for Hilbert spaces are interesting for mathematicians but also important for applications e.g. in signal analysis and in physics. Both in mathematics and physics it is natural to consider a full scale of spaces, and not only a single one. In this paper, we study how certain frame-related properties, as completeness or the property of being a (semi-)frame, of a certain sequence in one of the spaces propagate to other spaces in a scale. We link that to the properties of the respective frame-related operators, like analysis or synthesis. We start with a detailed survey of the theory of Hilbert chains. Using a canonical isomorphism the properties of frame sequences are naturally preserved between different spaces. We also show that some results can be transferred if the original sequence is considered, in particular that the upper semi-frame property is kept in larger spaces, while the lower one to smaller ones. This leads to a negative result: a sequence can never be a frame for two Hilbert spaces of the scale if the scale is non-trivial, i.e. spaces are not equal.

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arXiv2020

Weak $A$-frames and weak $A$-semi-frames

After reviewing the interplay between frames and lower semi-frames, we introduce the notion of lower semi-frame controlled by a densely defined operator $A$ or, for short, a \emph{weak lower $A$-semi-frame} and we study its properties. In particular, we compare it with that of lower atomic systems, introduced in \cite{GB}. We discuss duality properties and we suggest several possible definitions for weak $A$-upper semi-frames. Concrete examples are presented.

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arXiv2020

Frames and weak frames for unbounded operators

In 2012 Găvruţa introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$, in order to decompose its range $\mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:\mathcal{D}(A)\to\mathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the norm of $\mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the graph norm of $A$.

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arXiv2019

Continuous frames for unbounded operators

Few years ago Găvruţa gave the notions of $K$-frame and atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$ in order to decompose $\mathcal{R}(K)$, the range of $K$, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator on a Hilbert space $A$ in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.

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arXiv2017

Hamiltonians defined by biorthogonal sets

In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of $\mathcal{G}$-quasi basis and we show a series of conditions under which a definition of non self-adjoint Hamiltonian with purely point real spectra is still possible.

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arXiv2015

Riesz-like bases in rigged Hilbert spaces

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space $\D[t] \subset \H \subset \D^\times[t^\times]$. A Riesz-like basis, in particular, is obtained by considering a sequence $\{ξ_n\}\subset \D$ which is mapped by a one-to-one continuous operator $T:\D[t]\to\H[\|\cdot\|]$ into an orthonormal basis of the central Hilbert space $\H$ of the triplet. The operator $T$ is, in general, an unbounded operator in $\H$. If $T$ has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.

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arXiv2013

Extensions of Representable Positive Linear Functionals to Unitized Quasi *-Algebras

It is known that, under certain conditions, *-representability and extensibility to the unitized *-algebra of a positive linear functional, defined on a *-algebra without unit, are equivalent. In this paper, a new condition for an analogous result is given for the case of a hermitian linear functional defined on a quasi *-algebra $(\A,\Ao)$ without unit.

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arXiv2013

Rigged Hilbert spaces and contractive families of Hilbert spaces

The existence of a rigged Hilbert space whose extreme spaces are, respectively, the projective and the inductive limit of a directed contractive family of Hilbert spaces is investigated. It is proved that, when it exists, this rigged Hilbert space is the same as the canonical rigged Hilbert space associated to a family of closable operators in the central Hilbert space.

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arXiv2013

Bounded elements of C*-inductive locally convex spaces

The notion of bounded element of C*-inductive locally convex spaces (or C*-inductive partial *-algebras) is introduced and discussed in two ways: the first one takes into account the inductive structure provided by certain families of C*-algebras; the second one is linked to natural order of these spaces. A particular attention is devoted to the relevant instance provided by the space of continuous linear maps acting in a rigged Hilbert space.

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