Functional Models and Minimal Contractive Liftings

Based on a careful analysis of functional models for contractive multi-analytic operators we establish a one-to-one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi-analytic operators. This allows an effective construction and classification of all such liftings with given defects. Popescu's theory of characteristic functions of completely non-coisometric row contractions is obtained as a special case satisfying a Szeg\"{o} condition. In another special case of single contractions and defects equal to $1$ all non-zero Schur functions on the unit disk appear in the classification. It is also shown that the process of constructing liftings iteratively reflects itself in a factorization of the corresponding symbols.Comment: 25 page

Characteristic Functions of Liftings–II

We prove that the symbol of the characteristic function of a minimal contractive lifting is an injective map and that the converse also holds, using explicit computation and functional models. We discuss when the characteristic function of a lifting is a polynomial and present a series representation for the characteristic functions of liftings

Systemic effects of periodontitis treatment in patients with type 2 diabetes: a 12 month, single-centre, investigator-masked, randomised trial

BACKGROUND: Chronic inflammation is believed to be a major mechanism underlying the pathophysiology of type 2 diabetes. Periodontitis is a cause of systemic inflammation. We aimed to assess the effects of periodontal treatment on glycaemic control in people with type 2 diabetes. METHODS: In this 12 month, single-centre, parallel-group, investigator-masked, randomised trial, we recruited patients with type 2 diabetes, moderate-to-severe periodontitis, and at least 15 teeth from four local hospitals and 15 medical or dental practices in the UK. We randomly assigned patients (1:1) using a computer-generated table to receive intensive periodontal treatment (IPT; whole mouth subgingival scaling, surgical periodontal therapy [if the participants showed good oral hygiene practice; otherwise dental cleaning again], and supportive periodontal therapy every 3 months until completion of the study) or control periodontal treatment (CPT; supra-gingival scaling and polishing at the same timepoints as in the IPT group). Treatment allocation included a process of minimisation in terms of diabetes onset, smoking status, sex, and periodontitis severity. Allocation to treatment was concealed in an opaque envelope and revealed to the clinician on the day of first treatment. With the exception of dental staff who performed the treatment and clinical examinations, all study investigators were masked to group allocation. The primary outcome was between-group difference in HbA1c at 12 months in the intention-to-treat population. This study is registered with the ISRCTN registry, number ISRCTN83229304. FINDINGS: Between Oct 1, 2008, and Oct 31, 2012, we randomly assigned 264 patients to IPT (n=133) or CPT (n=131), all of whom were included in the intention-to-treat population. At baseline, mean HbA1c was 8·1% (SD 1·7) in both groups. After 12 months, unadjusted mean HbA1c was 8·3% (SE 0·2) in the CPT group and 7·8% (0·2) in the IPT group; with adjustment for baseline HbA1c, age, sex, ethnicity, smoking status, duration of diabetes, and BMI, HbA1c was 0·6% (95% CI 0·3-0·9; p<0·0001) lower in the IPT group than in the CPT group. At least one adverse event was reported in 30 (23%) of 133 patients in the IPT group and 23 (18%) of 131 patients in the CPT group. Serious adverse events were reported in 11 (8%) patients in the IPT group, including one (1%) death, and 11 (8%) patients in the CPT group, including three (2%) deaths. INTERPRETATION: Compared with CPT, IPT reduced HbA1c in patients with type 2 diabetes and moderate-to-severe periodontitis after 12 months. These results suggest that routine oral health assessment and treatment of periodontitis could be important for effective management of type 2 diabetes. FUNDING: Diabetes UK and UK National Institute for Health Research.Diabetes UK and UK National Institute for Health Researc

Commuting row contractions with polynomial characteristic functions

A characteristic function is a special operator-valued analytic function defined on the open unit ball of C n associated with an n-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of n-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason’s problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions