Operators in Rigged Hilbert spaces: some spectral properties

A notion of resolvent set for an operator acting in a rigged Hilbert space \D \subset \H\subset \D^\times is proposed. This set depends on a family of intermediate locally convex spaces living between \D and \D^\times, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.Comment: 29 page

Fully representable and *-semisimple topological partial *-algebras

We continue our study of topological partial *-algebras, focusing our attention to *-semisimple partial *-algebras, that is, those that possess a {multiplication core} and sufficiently many *-representations. We discuss the respective roles of invariant positive sesquilinear (ips) forms and representable continuous linear functionals and focus on the case where the two notions are completely interchangeable (fully representable partial *-algebras) with the scope of characterizing a *-semisimple partial *-algebra. Finally we describe various notions of bounded elements in such a partial *-algebra, in particular, those defined in terms of a positive cone (order bounded elements). The outcome is that, for an appropriate order relation, one recovers the \M-bounded elements introduced in previous works.Comment: 26 pages, Studia Mathematica (2012) to appea

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

Quasi *-algebras possessing a sufficient family M of invariant positive sesquilinear forms carry several topologies related to 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 M, is a dense C*-algebra

Disequazioni variazionali, problemi di ottimo e convessita' generalizzata

Il lavoro fatto nella tesi propone un'estensione dei principali teoremi sulle disequazioni variazionali vettoriali (VVI), considerando l'ordinamento indotto da un generico cono C di R^p (convesso, chiuso, puntato e con interno non vuoto) anziche' dal classico cono d'ordine R^p_+ . Uno dei principali risultati e' l'estensione di un risultato in [2], che lega le VVI ad una famiglia di disequazioni variazionali scalari dipendenti da un parametro. Da esso seguono alcuni dei principali teoremi di esistenza mediante il riconducimento al caso scalare. Il risultato, inoltre, insieme ad opportune ipotesi di C-convessita' generalizzata sulla funzione obiettivo, garantisce l'esistenza di soluzioni per problemi di ottimo vettoriale. Il lavoro si conclude con le dimostrazioni di due nuove condizioni sufficienti di buona posizione per una disequazione variazionale vettoriale debole (nel seguito VVI^w) il cui operatore ammette primitiva. La prima amplia la classe delle funzioni per cui viene garantita la buona posizione di VVI^w(X, Jf ), facendo uso dell'ipotesi di C-pseudo- convessit'a per la funzione f , anziche' di quella di C-convessit'a. L'altra condizione, oltre a far uso di quest'ipotesi piu' debole su f , richiede la connessione di alcuni insiemi $G_{c^0}(\epsilon$ coinvolti nella definizione di buona posizione e la limitatezza dell'insieme delle soluzioni del problema di ottimo vettoriale debole (nel seguito VOP^w). Da tali risultati sono poi dedotte altrettante condizioni sufficienti per la buona posizione di VOP^w( f , X)

Frame-Related Sequences in Chains and Scales of Hilbert Spaces

Frames for Hilbert spaces are interesting for mathematicians but also important for applications in, e.g., signal analysis and physics. In both 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 of a certain sequence in one of the spaces, such as completeness or the property of being a (semi-) frame, propagate to the other ones in a scale of Hilbert spaces. We link that to the properties of the respective frame-related operators, such as 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 is kept in 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., if the spaces are not equal

Pedagogical Models Of Surface Mechanical Wave Propagation In Various Materials

We report on a teaching approach oriented to the understanding of some relevant concepts of wave propagation in solids. It is based on simple experiments involving the propagation of shock mechanical waves in solid slabs of various materials. Methods similar to the generation and propagation of seismic waves are adopted. Educational seismometers, interfaced with computers, are used to detect and visualize the shock waves and to analyse their propagation properties. A qualitative discussion of the results concerning the propagation and the attenuation of the waves allows us to draw basic conclusions about the response of the matter to solicitation impacts and their propagation

Spatially restricted expression of PlOtp, a Paracentrotus lividus Orthopedia-related homeobox gene, is correlated with oral ectodermal patterning and skeletal morphogenesis in late-cleavage sea urchin embryos

Several homeobox genes are expressed in the sea urchin embryo but their roles in development have yet to be elucidated. Of particular interest are homologues of homeobox genes that in mouse and Drosophila are involved in patterning the developing central nervous system (CNS). Here, we report the cloning of an orthpedia (Otp)-related gene from Paracentrotus lividus, PlOtp. Otp is a single copy zygotic gene that presents a unique and highly restricted expression pattern. Transcripts were first detected at the mid-gastrula stage in two pairs of oral ectoderm cells located in a ventrolateral position, overlying primary mesenchyme cell (PMC) clusters. Increases in both transcript abundance and the number of Otp-expressing cells were observed at prism and pluteus stages. Otp transcripts are symmetrically distributed in a few ectodermal cells of the oral field. Labelled cells were observed close to sites of active skeletal rod growth (tips of the budding oral and anal arms), and at the juxtaposition of stomodeum and foregut. Chemicals known to perturb PMC patterning along animal-vegetal and oral-aboral axes altered the pattern of Otp expression. Vegetalization by LiCl caused a shift in Otp-expressing cells toward the animal pole, adjacent to shifted PMC aggregates. Nickel treatment induced expression of the Otp gene in an increased number of ectodermal cells, which adopted a radialized pattern. Finally, ectopic expression of Otp mRNA affected patterning along the oral-aboral axis and caused skeletal abnormalities that resembled those exhibited by nickel-treated embryos. From these results, we conclude that the Otp homeodomain gene is involved in short-range cell signalling within the oral ectoderm for patterning the endoskeleton of the larva through epithelial-mesenchymal interactions

Supplemento ai Rendiconti del Circolo Matematico di Palermo

In this paper we show how some known quasi *-algebras can also be obtained through the construction of slight extensions of nonclosable positive linear functionals defined on dense *-subalgebras Ao of given topological *-algebras. Moreover, we consider also, for each of the functionals we present, their absolutely convergent extensions and the GNS *-representations of the quasi *-algebras arising in the process of extension

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

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 H, in order to decompose its range R(K) with a frame-like expansion. In this article, we revisit these concepts for an unbounded and densely defined operator A: D(A) → H in two different ways. In one case, we consider a non-Bessel sequence where the coefficient sequence depends continuously on f∈ D(A) with respect to the norm of H. In the other case, we consider a Bessel sequence and the coefficient sequence depends continuously on f∈ D(A) with respect to the graph norm of A