Applications of Hilbert Module Approach to Multivariable Operator Theory

A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space \clh associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: $$\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,$$where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and geometric aspects of Hilbert module approach to operator theory and to survey several applications of the theory of Hilbert modules in multivariable operator theory. The topics which are studied include: generalized canonical models and Cowen-Douglas class, dilations and factorization of reproducing kernel Hilbert spaces, a class of simple submodules and quotient modules of the Hardy modules over polydisc, commutant lifting theorem, similarity and free Hilbert modules, left invertible multipliers, inner resolutions, essentially normal Hilbert modules, localizations of free resolutions and rigidity phenomenon. This article is a companion paper to "An Introduction to Hilbert Module Approach to Multivariable Operator Theory".Comment: 46 pages. This is a companion paper to arXiv:1308.6103. To appear in Handbook of Operator Theory, Springe

Involvement in the US criminal justice system and cost implications for persons treated for schizophrenia

<p>Abstract</p> <p>Background</p> <p>Individuals with schizophrenia may have a higher risk of encounters with the criminal justice system than the general population, but there are limited data on such encounters and their attendant costs. This study assessed the prevalence of encounters with the criminal justice system, encounter types, and the estimated cost attributable to these encounters in the one-year treatment of persons with schizophrenia.</p> <p>Methods</p> <p>This post-hoc analysis used data from a prospective one-year cost-effectiveness study of persons treated with antipsychotics for schizophrenia and related disorders in the United States. Criminal justice system involvement was assessed using the Schizophrenia Patients Outcome Research Team (PORT) client survey and the victimization subscale of the Lehman Quality of Life Interview (QOLI). Direct cost of criminal justice system involvement was estimated using previously reported costs per type of encounter. Patients with and without involvement were compared on baseline characteristics and direct annual health care and criminal justice system-related costs.</p> <p>Results</p> <p>Overall, 278 (46%) of 609 participants reported at least 1 criminal justice system encounter. They were more likely to be substance users and less adherent to antipsychotics compared to participants without involvement. The 2 most prevalent types of encounters were being a victim of a crime (67%) and being on parole or probation (26%). The mean annual per-patient cost of involvement was $1,429, translating to 6% of total annual direct health care costs for those with involvement (11% when excluding crime victims).</p> <p>Conclusions</p> <p>Criminal justice system involvement appears to be prevalent and costly for persons treated for schizophrenia in the United States. Findings highlight the need to better understand the interface between the mental health and the criminal justice systems and the related costs, in personal, societal, and economic terms.</p

The Deficit Gamble

The historical behavior of interest rates and growth rates in U.S. data suggests that the government can, with a high probability, run temporary budget deficits and then roll over the resulting government debt forever. The purpose of this paper is to document this finding and to examine its implications. Using a standard overlapping-generations model of capital accumulation, we show that whenever a perpetual rollover of debt succeeds, policy can make every generation better off. This conclusion does not imply that deficits are good policy, for an attempt to roll over debt forever might fail. But the adverse effects of deficits, rather than being inevitable, occur with only a small probability.

Operator theory and function theory in Drury-Arveson space and its quotients

The Drury-Arveson space $H^2_d$, also known as symmetric Fock space or the $d$-shift space, is a Hilbert function space that has a natural $d$-tuple of operators acting on it, which gives it the structure of a Hilbert module. This survey aims to introduce the Drury-Arveson space, to give a panoramic view of the main operator theoretic and function theoretic aspects of this space, and to describe the universal role that it plays in multivariable operator theory and in Pick interpolation theory.Comment: Final version (to appear in Handbook of Operator Theory); 42 page

Progress in noncommutative function theory

In this expository paper we describe the study of certain non-self-adjoint operator algebras, the Hardy algebras, and their representation theory. We view these algebras as algebras of (operator valued) functions on their spaces of representations. We will show that these spaces of representations can be parameterized as unit balls of certain $W^{*}$-correspondences and the functions can be viewed as Schur class operator functions on these balls. We will provide evidence to show that the elements in these (non commutative) Hardy algebras behave very much like bounded analytic functions and the study of these algebras should be viewed as noncommutative function theory

Characterisation and expression of SPLUNC2, the human orthologue of rodent parotid secretory protein

We recently described the Palate Lung Nasal Clone (PLUNC) family of proteins as an extended group of proteins expressed in the upper airways, nose and mouth. Little is known about these proteins, but they are secreted into the airway and nasal lining fluids and saliva where, due to their structural similarity with lipopolysaccharide-binding protein and bactericidal/permeability-increasing protein, they may play a role in the innate immune defence. We now describe the generation and characterisation of novel affinity-purified antibodies to SPLUNC2, and use them to determine the expression of this, the major salivary gland PLUNC. Western blotting showed that the antibodies identified a number of distinct protein bands in saliva, whilst immunohistochemical analysis demonstrated protein expression in serous cells of the major salivary glands and in the ductal lumens as well as in cells of minor mucosal glands. Antibodies directed against distinct epitopes of the protein yielded different staining patterns in both minor and major salivary glands. Using RT-PCR of tissues from the oral cavity, coupled with EST analysis, we showed that the gene undergoes alternative splicing using two 5' non-coding exons, suggesting that the gene is regulated by alternative promoters. Comprehensive RACE analysis using salivary gland RNA as template failed to identify any additional exons. Analysis of saliva showed that SPLUNC2 is subject to N-glycosylation. Thus, our study shows that multiple SPLUNC2 isoforms are found in the oral cavity and suggest that these proteins may be differentially regulated in distinct tissues where they may function in the innate immune response

Succeeding against the odds: can schools ‘compensate for society’?

Education researchers, policy-makers and practitioners in the UK have debated the question of what, and how much, schools can do to mitigate the effects of parental background on educational outcomes over the last half a century. A range of programmes, strategies and interventions have been implemented, and continue to be implemented in an effort to ‘break the link’ between socio-economic disadvantage and low educational outcomes, but educational inequalities have persisted. This paper draws on theoretical and empirical research to offer a new analysis of compensatory education in England across three main phases since the 1960s

Flow in Open Channel with Complex Solid Boundary

yesA two-dimensional steady potential flow theory is applied to calculate the flow in an open channel with complex solid boundaries. The boundary integral equations for the problem under investigation are first derived in an auxiliary plane by taking the Cauchy integral principal values. To overcome the difficulties of a nonlinear curvilinear solid boundary character and free water surface not being known a priori, the boundary integral equations are transformed to the physical plane by substituting the integral variables. As such, the proposed approach has the following advantages: (1) the angle of the curvilinear solid boundary as well as the location of free water surface (initially assumed) is a known function of coordinates in physical plane; and (2) the meshes can be flexibly assigned on the solid and free water surface boundaries along which the integration is performed. This avoids the difficulty of the traditional potential flow theory, which seeks a function to conformally map the geometry in physical plane onto an auxiliary plane. Furthermore, rough bed friction-induced energy loss is estimated using the Darcy-Weisbach equation and is solved together with the boundary integral equations using the proposed iterative method. The method has no stringent requirement for initial free-water surface position, while traditional potential flow methods usually have strict requirement for the initial free-surface profiles to ensure that the numerical computation is stable and convergent. Several typical open-channel flows have been calculated with high accuracy and limited computational time, indicating that the proposed method has general suitability for open-channel flows with complex geometry