Toeplitz operators on finite and infinite dimensional spaces with associated psi *-Fréchet algebras

The present thesis is a contribution to the multi-variable theory of Bergman and Hardy Toeplitz operators on spaces of holomorphic functions over finite and infinite dimensional domains. In particular, we focus on certain spectral invariant Frechet operator algebras F closely related to the local symbol behavior of Toeplitz operators in F. We summarize results due to B. Gramsch et.al. on the construction of Psi_0- and Psi^*-algebras in operator algebras and corresponding scales of generalized Sobolev spaces using commutator methods, generalized Laplacians and strongly continuous group actions. In the case of the Segal-Bargmann space H^2(C^n,m) of Gaussian square integrable entire functions on C^n we determine a class of vector-fields Y(C^n) supported in complex cones K. Further, we require that for any finite subset V of Y(C^n) the Toeplitz projection P is a smooth element in the Psi_0-algebra constructed by commutator methods with respect to V. As a result we obtain Psi_0- and Psi^*-operator algebras F localized in cones K. It is an immediate consequence that F contains all Toeplitz operators T_f with a symbol f of certain regularity in an open neighborhood of K. There is a natural unitary group action on H^2(C^n,m) which is induced by weighted shifts and unitary groups on C^n. We examine the corresponding Psi^*-algebra A of smooth elements in Toeplitz-C^*-algebras. Among other results sufficient conditions on the symbol f for T_f to belong to A are given in terms of estimates on its Berezin-transform. Local aspects of the Szegö projection P_s on the Heisenbeg group and the corresponding Toeplitz operators T_f with symbol f are studied. In this connection we apply a result due to Nagel and Stein which states that for any strictly pseudo-convex domain U the projection P_s is a pseudodifferential operator of exotic type (1/2, 1/2). The second part of this thesis is devoted to the infinite dimensional theory of Bergman and Hardy spaces and the corresponding Toeplitz operators. We give a new proof of a result observed by Boland and Waelbroeck. Namely, that the space of all holomorphic functions H(U) on an open subset U of a DFN-space (dual Frechet nuclear space) is a FN-space (Frechet nuclear space) equipped with the compact open topology. Using the nuclearity of H(U) we obtain Cauchy-Weil-type integral formulas for closed subalgebras A in H_b(U), the space of all bounded holomorphic functions on U, where A separates points. Further, we prove the existence of Hardy spaces of holomorphic functions on U corresponding to the abstract Shilov boundary S_A of A and with respect to a suitable boundary measure on S_A. Finally, for a domain U in a DFN-space or a polish spaces we consider the symmetrizations m_s of measures m on U by suitable representations of a group G in the group of homeomorphisms on U. In particular,in the case where m leads to Bergman spaces of holomorphic functions on U, the group G is compact and the representation is continuous we show that m_s defines a Bergman space of holomorphic functions on U as well. This leads to unitary group representations of G on L^p- and Bergman spaces inducing operator algebras of smooth elements related to the symmetries of U.

Pseudodifferential analysis in Psi*-algebras on transmission spaces, infinite solving ideal chains and K-theory for conformally compact spaces

The present thesis is a contribution to the theory of algebras of pseudodifferential operators on singular settings. In particular, we focus on the $b$-calculus and the calculus on conformally compact spaces in the sense of Mazzeo and Melrose in connection with the notion of spectral invariant transmission operator algebras. We summarize results given by Gramsch et. al. on the construction of $Psi_0$-and $Psi*$-algebras and the corresponding scales of generalized Sobolev spaces using commutators of certain closed operators and derivations. In the case of a manifold with corners $Z$ we construct a $Psi*$-completion $A_b(Z,{}^bOmega^{1/2})$ of the algebra of zero order $b$-pseudodifferential operators $Psi_{b,cl}(Z, {}^bOmega^{1/2})$ in the corresponding $C*$-closure $B(Z,{}^bOmega^{12})hookrightarrow L(L^2(Z,{}^bOmega^{1/2}))$. The construction will also provide that localised to the (smooth) interior of Z the operators in the $A_b(Z, {}^bOmega^{1/2})$ can be represented as ordinary pseudodifferential operators. In connection with the notion of solvable $C*$-algebras - introduced by Dynin - we calculate the length of the $C*$-closure of $Psi_{b,cl}^0(F,{}^bOmega^{1/2},R^{E(F)})$ in $B(F,{}^bOmega^{1/2}),R^{E(F)})$ by localizing $B(Z, {}^bOmega^{1/2})$ along the boundary face $F$ using the (extended) indical familiy $I^B_{FZ}$. Moreover, we discuss how one can localise a certain solving ideal chain of $B(Z, {}^bOmega^{1/2})$ in neighbourhoods $U_p$ of arbitrary points $pin Z$. This localisation process will recover the singular structure of $U_p$; further, the induced length function $l_p$ is shown to be upper semi-continuous. We give construction methods for $Psi*$- and $C*$-algebras admitting only infinite long solving ideal chains. These algebras will first be realized as unconnected direct sums of (solvable) $C*$-algebras and then refined such that the resulting algebras have arcwise connected spaces of one dimensional representations. In addition, we recall the notion of transmission algebras on manifolds with corners $(Z_i)_{iin N}$ following an idea of Ali Mehmeti, Gramsch et. al. Thereby, we connect the underlying $C^infty$-function spaces using point evaluations in the smooth parts of the $Z_i$ and use generalized Laplacians to generate an appropriate scale of Sobolev spaces. Moreover, it is possible to associate generalized (solving) ideal chains to these algebras, such that to every $ninN$ there exists an ideal chain of length $n$ within the algebra. Finally, we discuss the $K$-theory for algebras of pseudodifferential operators on conformally compact manifolds $X$ and give an index theorem for these operators. In addition, we prove that the Dirac-operator associated to the metric of a conformally compact manifold $X$ is not a Fredholm operator.

Weighted Inequalities and Lipschitz Spaces

In this thesis I have characterized the trace measures for particular potential spaces of functions defined on R^n, but "mollified" so that the potentials are de facto defined on the upper half-space of R^n. The potential functions are kind Riesz-Bessel. The characterization of trace measures for these spaces is a test condition on elementary sets of the upper half-space. To prove the test condition as sufficient condition for trace measures, I had give an extension to the case of upper half-space of the Muckenhoupt-Wheeden and Wolff inequalities. Finally I characterized the Carleson-trace measures for Besov spaces of discrete martingales. This is a simplified discrete model for harmonic extensions of Lipschitz-Besov spaces.

The Universe 10 Billion years ago: Morphologies, Star-formation rates and Galactic-scale winds in z ∼ 2 Galaxies

This thesis is devoted to the study of the properties of high-redsfhit galaxies in the epoch 1 < z < 3, when a substantial fraction of galaxy mass was assembled, and when the evolution of the star-formation rate density peaked. Following a multi-perspective approach and using the most recent and high-quality data available (spectra, photometry and imaging), the morphologies and the star-formation properties of high-redsfhit galaxies were investigated. Through an accurate morphological analyses, the built up of the Hubble sequence was placed around z ~ 2.5. High-redshift galaxies appear, in general, much more irregular and asymmetric than local ones. Moreover, the occurrence of morphological k-­correction is less pronounced than in the local Universe. Different star-formation rate indicators were also studied. The comparison of ultra-violet and optical based estimates, with the values derived from infra-red luminosity showed that the traditional way of addressing the dust obscuration is problematic, at high-redshifts, and new models of dust geometry and composition are required. Finally, by means of stacking techniques applied to rest-frame ultra-violet spectra of star-forming galaxies at z~2, the warm phase of galactic-scale outflows was studied. Evidence was found of escaping gas at velocities of ~ 100 km/s. Studying the correlation of inter-­stellar absorption lines equivalent widths with galaxy physical properties, the intensity of the outflow-related spectral features was proven to depend strongly on a combination of the velocity dispersion of the gas and its geometry.

On moduli spaces of semistable sheaves on K3 surfaces

Ich untersuche die nicht bereits durch die Arbeit "Singular symplectic moduli spaces" von Kaledin, Lehn und Sorger (Invent. Math. 164 (2006), no. 3) abgedeckten Fälle von Modulräumen halbstabiler Garben auf projektiven K3-Flächen - die Fälle mit Mukai-Vektor (0,c,0) sowie die Modulräume zu nichtgenerischen amplen Divisoren - hinsichtlich der möglichen Konstruktion neuer Beispiele von kompakten irreduziblen symplektischen Mannigfaltigkeiten. Ich stelle einen Zusammenhang zu den bereits untersuchten Modulräumen und Verallgemeinerungen derselben her und erweitere bekannte Ergebnisse auf alle offenen Fälle von Garben vom Rang 0 und viele Fälle von Garben von positivem Rang. Insbesondere kann in diesen Fällen die Existenz neuer Beispiele von kompakten irreduziblen symplektischen Mannigfaltigkeiten, die birational über Komponenten des Modulraums liegen, ausgeschlossen werden.

The evolution of massive clumps in star forming regions

In this thesis two related arguments are investigated: - The first stages of the process of massive star formation, investigating the physical conditions and -properties of massive clumps in different evolutionary stages, and their CO depletion; - The influence that high-mass stars have on the nearby material and on the activity of star formation. I characterise the gas and dust temperature, mass and density of a sample of massive clumps, and analyse the variation of these properties from quiescent clumps, without any sign of active star formation, to clumps likely hosting a zero-age main sequence star. I briefly discuss CO depletion and recent observations of several molecular species, tracers of Hot Cores and/or shocked gas, of a subsample of these clumps. The issue of CO depletion is addressed in more detail in a larger sample consisting of the brightest sources in the ATLASGAL survey: using a radiative tranfer code I investigate how the depletion changes from dark clouds to more evolved objects, and compare its evolution to what happens in the low-mass regime. Finally, I derive the physical properties of the molecular gas in the photon-dominated region adjacent to the HII region G353.2+0.9 in the vicinity of Pismis 24, a young, massive cluster, containing some of the most massive and hottest stars known in our Galaxy. I derive the IMF of the cluster and study the star formation activity in its surroundings. Much of the data analysis is done with a Bayesian approach. Therefore, a separate chapter is dedicated to the concepts of Bayesian statistics.

Knots and links in lens spaces

The aim of this dissertation is to improve the knowledge of knots and links in lens spaces. If the lens space L(p,q) is defined as a 3-ball with suitable boundary identifications, then a link in L(p,q) can be represented by a disk diagram, i.e. a regular projection of the link on a disk. In this contest, we obtain a complete finite set of Reidemeister-type moves establishing equivalence, up to ambient isotopy. Moreover, the connections of this new diagram with both grid and band diagrams for links in lens spaces are shown. A Wirtinger-type presentation for the group of the link and a diagrammatic method giving the first homology group are described. A class of twisted Alexander polynomials for links in lens spaces is computed, showing its correlation with Reidemeister torsion. One of the most important geometric invariants of links in lens spaces is the lift in 3-sphere of a link L in L(p,q), that is the counterimage of L under the universal covering of L(p,q). Starting from the disk diagram of the link, we obtain a diagram of the lift in the 3-sphere. Using this construction it is possible to find different knots and links in L(p,q) having equivalent lifts, hence we cannot distinguish different links in lens spaces only from their lift. The two final chapters investigate whether several existing invariants for links in lens spaces are essential, i.e. whether they may assume different values on links with equivalent lift. Namely, we consider the fundamental quandle, the group of the link, the twisted Alexander polynomials, the Kauffman Bracket Skein Module and an HOMFLY-PT-type invariant.

The core mass function in star-forming region NGC6357

The aim of this work was to study the dense cloud structures and to obtain the mass distribution of the dense cores (CMF) within the NGC6357 complex, from observations of the dust continuum at 450 and 850~$\mu$m of a 30 $\times$ 30 arcmin$^2$ region containing the H\textsc{ii} regions, G353.2+0.9 and G353.1+0.6.

Moduli spaces of (G,h)-constellations

Given a reductive group G acting on an affine scheme X over C and a Hilbert function h: Irr G → N_0, we construct the moduli space M_Ө(X) of Ө-stable (G,h)-constellations on X, which is a common generalisation of the invariant Hilbert scheme after Alexeev and Brion and the moduli space of Ө-stable G-constellations for finite groups G introduced by Craw and Ishii. Our construction of a morphism M_Ө(X) → X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G. Furthermore, we determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C²)⁶ as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for the realisation of such calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.

Higher order Sobolev Spaces and polyharmonic boundary value problems in Carnot Groups

The main task of this work is to present a concise survey on the theory of certain function spaces in the contexts of Hörmander vector fields and Carnot Groups, and to discuss briefly an application to some polyharmonic boundary value problems on Carnot Groups of step 2.

Tracing AGN accretion and star formation in Herschel galaxies

In this Thesis, we study the physical properties and the cosmic evolution of AGN and their host galaxies since z∼3. Our analysis exploits samples of star forming galaxies detected with Herschel at far-IR wavelengths (from 70 up to 500 micron) in different extragalactic surveys, such as COSMOS and the deep GOODS (South and North) fields. The broad-band ancillary data available in COSMOS and the GOODS fields, allows us to implement Herschel and Spitzer photometry with multi-wavelength ancillary data. We perform a multicomponent SED-fitting decomposition to decouple the emission due to star formation from that due to AGN accretion, and to estimate both host-galaxy parameters (such as stellar mass, M* and star formation rate, SFR), and nuclear intrinsic bolometric luminosities. We use the individual estimates of AGN bolometric luminosity obtained through SED-fitting decomposition to reconstruct the redshit evolution of the AGN bolometric luminosity function since z∼3. The resulting trends are used to estimate the overall AGN accretion rate density at different cosmic epochs and to trace the first ever estimate of the AGN accretion history from an IR survey. Later on, we focus our study on the connection between AGN accretion and integrated galaxy properties. We analyse the relationships of AGN accretion with galaxy properties in the SFR-M* plane and at different cosmic epochs. Finally, we infer what is the parameter that best correlates with AGN accretion, comparing our results with previous studies and discussing their physical implications in the context of current scenarios of AGN/galaxy evolution.

Fibered knots and links in lens spaces

La tesi propone alcuni esempi di link fibrati in spazi lenticolari. Sfruttando la compatibilità fra le mosse di chirurgia intera e la nozione di open book decomposition, si ricava un esempio di link fibrato prima in L(p,1), per poi generalizzarlo a L(p,q). Si conclude determinando una struttura di contatto equivalente alla open book relativa agli spazi del tipo L(p,1).

Cooperative Cognitive Wireless Networks over TV White and Gray Spaces

Wireless networks rapidly became a fundamental pillar of everyday activities. Whether at work or elsewhere, people often benefits from always-on connections. This trend is likely to increase, and hence actual technologies struggle to cope with the increase in traffic demand. To this end, Cognitive Wireless Networks have been studied. These networks aim at a better utilization of the spectrum, by understanding the environment in which they operate, and adapt accordingly. In particular recently national regulators opened up consultations on the opportunistic use of the TV bands, which became partially free due to the digital TV switch over. In this work, we focus on the indoor use of of TVWS. Interesting use cases like smart metering and WiFI like connectivity arise, and are studied and compared against state of the art technology. New measurements for TVWS networks will be presented and evaluated, and fundamental characteristics of the signal derived. Then, building on that, a new model of spectrum sharing, which takes into account also the height from the terrain, is presented and evaluated in a real scenario. The principal limits and performance of TVWS operated networks will be studied for two main use cases, namely Machine to Machine communication and for wireless sensor networks, particularly for the smart grid scenario. The outcome is that TVWS are certainly interesting to be studied and deployed, in particular when used as an additional offload for other wireless technologies. Seeing TVWS as the only wireless technology on a device is harder to be seen: the uncertainity in channel availability is the major drawback of opportunistic networks, since depending on the primary network channel allocation might lead in having no channels available for communication. TVWS can be effectively exploited as offloading solutions, and most of the contributions presented in this work proceed in this direction.

Recycling the city: a sustainable planning framework to reduce, reuse and recycle urban residual spaces

Cities are key locations where Sustainability needs to be addressed at all levels, as land is a finite resource. However, not all urban spaces are exploited at best, and land developers often evaluate unused, misused, or poorly-designed urban portions as impracticable constraints. Further, public authorities lose the challenge to enable and turn these urban spaces into valuable opportunities where Sustainable Urban Development may flourish. Arguing that these spatial elements are at the centre of SUD, the paper elaborates a prototype in the form of a conceptual strategic planning framework, committed to an effective recycling of the city spaces using a flexible and multidisciplinary approach. Firstly, the research focuses upon a broad review of Sustainability literature, highlighting established principles and guidelines, building a sound theoretical base for the new concept. Hence, it investigates origins, identifies and congruently suggests a definition, characterisation and classification for urban “R-Spaces”. Secondly, formal, informal and temporary fitting functions are analysed and inserted into a portfolio meant to enhance adaptability and enlarge the choices for the on-site interventions. Thirdly, the study outlines ideal quality requirements for a sustainable planning process. Then, findings are condensed in the proposal, which is articulated in the individuation of tools, actors, plans, processes and strategies. Afterwards, the prototype is tested upon case studies: Solar Community (Casalecchio di Reno, Bologna) and Hyllie Sustainable City Project, the latter developed via an international workshop (ACSI-Camp, Malmö, Sweden). Besides, the qualitative results suggest, inter alia, the need to right-size spatial interventions, separate structural and operative actors, involve synergies’ multipliers and intermediaries (e.g. entrepreneurial HUBs, innovation agencies, cluster organisations…), maintain stakeholders’ diversity and create a circular process open for new participants. Finally, the paper speculates upon a transfer of the Swedish case study to Italy, and then indicates desirable future researches to favour the prototype implementation.

T2 star relaxation times for assessment of articular cartilage at 3 T: a feasibility study

T2 mapping techniques use the relaxation constant as an indirect marker of cartilage structure, and the relaxation constant has also been shown to be a sensitive parameter for cartilage evaluation. As a possible additional robust biomarker, T2* relaxation time is a potential, clinically feasible parameter for the biochemical evaluation of articular cartilage.