Small eigenvalues of large Hermitian moment matrices

We consider an infinite Hermitian positive definite matrix M which is the moment matrix associated with a measure μ with infinite and compact support on the complex plane. We prove that if the polynomials are dense in L2(μ) then the smallest eigenvalue λn of the truncated matrix Mn of M of size (n+1)×(n+1) tends to zero when n tends to infinity. In the case of measures in the closed unit disk we obtain some related results

A characterization of polynomial density on curves via matrix algebra

In this work, our aim is to obtain conditions to assure polynomial approximation in Hilbert spaces L2(m), with m a compactly supported measure in the complex plane, in terms of properties of the associated moment matrix with the measure m. To do it, in the more general context of Hermitian positive semidefinite matrices, we introduce two indexes, g(M) and l(M), associated with different optimization problems concerning theses matrices. Our main result is a characterization of density of polynomials in the case of measures supported on Jordan curves with non-empty interior using the index g and other specific index related to it. Moreover, we provide a new point of view of bounded point evaluations associated with a measure in terms of the index g that will allow us to give an alternative proof of Thomson's theorem, by using these matrix indexes. We point out that our techniques are based in matrix algebra tools in the framework of Hermitian positive definite matrices and in the computation of certain indexes related to some optimization problems for infinite matrices

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

We characterize Hessenberg matrices D associated with measures in the unit circle ν, which are matrix representations of compact and actually Hilbert Schmidt perturbations of the forward shift operator as those with recursion coefficients urn:x wiley:mma:media:mma5716:mma5716-math-0001 verifying urn:x-wiley:mma:media:mma5716:mma5716-math-0002, ie, associated with measures verifying Szegö condition. As a consequence, we obtain the following dichotomy result for Hessenberg matrices associated with measures in the unit circle: either D=SR+K2 with K2, a Hilbert Schmidt matrix, or there exists an unitary matrix U and a diagonal matrix Λ such that urn:x-wiley:mma:media:mma5716:mma5716-math-0003 with K2, a Hilbert Schmidt matrix. Moreover, we prove that for 1 ≤ p ≤ 2, if urn:x-wiley:mma:media:mma5716:mma5716-math-0004, then D=SR+Kp with Kp an absolutely p summable matrix inducing an operator in the p Schatten class. Some applications are given to classify measures on the unit circle

Evaluation of a quality improvement intervention to reduce anastomotic leak following right colectomy (EAGLE): pragmatic, batched stepped-wedge, cluster-randomized trial in 64 countries

Background Anastomotic leak affects 8 per cent of patients after right colectomy with a 10-fold increased risk of postoperative death. The EAGLE study aimed to develop and test whether an international, standardized quality improvement intervention could reduce anastomotic leaks. Methods The internationally intended protocol, iteratively co-developed by a multistage Delphi process, comprised an online educational module introducing risk stratification, an intraoperative checklist, and harmonized surgical techniques. Clusters (hospital teams) were randomized to one of three arms with varied sequences of intervention/data collection by a derived stepped-wedge batch design (at least 18 hospital teams per batch). Patients were blinded to the study allocation. Low- and middle-income country enrolment was encouraged. The primary outcome (assessed by intention to treat) was anastomotic leak rate, and subgroup analyses by module completion (at least 80 per cent of surgeons, high engagement; less than 50 per cent, low engagement) were preplanned. Results A total 355 hospital teams registered, with 332 from 64 countries (39.2 per cent low and middle income) included in the final analysis. The online modules were completed by half of the surgeons (2143 of 4411). The primary analysis included 3039 of the 3268 patients recruited (206 patients had no anastomosis and 23 were lost to follow-up), with anastomotic leaks arising before and after the intervention in 10.1 and 9.6 per cent respectively (adjusted OR 0.87, 95 per cent c.i. 0.59 to 1.30; P = 0.498). The proportion of surgeons completing the educational modules was an influence: the leak rate decreased from 12.2 per cent (61 of 500) before intervention to 5.1 per cent (24 of 473) after intervention in high-engagement centres (adjusted OR 0.36, 0.20 to 0.64; P < 0.001), but this was not observed in low-engagement hospitals (8.3 per cent (59 of 714) and 13.8 per cent (61 of 443) respectively; adjusted OR 2.09, 1.31 to 3.31). Conclusion Completion of globally available digital training by engaged teams can alter anastomotic leak rates. Registration number: NCT04270721 (http://www.clinicaltrials.gov)