Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.Comment: 63 pages, 11 figure

Safety of hospital discharge before return of bowel function after elective colorectal surgery

Background: Ileus is common after colorectal surgery and is associated with an increased risk of postoperative complications. Identifying features of normal bowel recovery and the appropriateness for hospital discharge is challenging. This study explored the safety of hospital discharge before the return of bowel function.Methods: A prospective, multicentre cohort study was undertaken across an international collaborative network. Adult patients undergoing elective colorectal resection between January and April 2018 were included. The main outcome of interest was readmission to hospital within 30 days of surgery. The impact of discharge timing according to the return of bowel function was explored using multivariable regression analysis. Other outcomes were postoperative complications within 30 days of surgery, measured using the Clavien-Dindo classification system.Results: A total of 3288 patients were included in the analysis, of whom 301 (9.2 per cent) were discharged before the return of bowel function. The median duration of hospital stay for patients discharged before and after return of bowel function was 5 (i.q.r. 4-7) and 7 (6-8) days respectively (P < 0.001). There were no significant differences in rates of readmission between these groups (6.6 versus 8.0 per cent; P = 0.499), and this remained the case after multivariable adjustment for baseline differences (odds ratio 0.90, 95 per cent c.i. 0.55 to 1.46; P = 0.659). Rates of postoperative complications were also similar in those discharged before versus after return of bowel function (minor: 34.7 versus 39.5 per cent; major 3.3 versus 3.4 per cent; P = 0.110).Conclusion: Discharge before return of bowel function after elective colorectal surgery appears to be safe in appropriately selected patients