Geometric properties and continuity of the pre-duality mapping in Banach space

We use the preduality mapping in proving characterizations of some geometric properties of Banach spaces. In particular, those include nearly strongly convexity, nearly uniform convexity-a property introduced by K. Goebel and T. Sekowski-, and nearly very convexity.We thank a referee for the careful reading of the manuscript. His/her observations substantially improved the overall aspect of the present work, detected several misprints and made some convenient changes. This work was supported by: (1) The National Natural Science Foundation of China (Grant no. 11271248). (2) Specific Academic Discipline Project of Shanghai Municipal Education Commission (Grant no. B-8932-13-0136). (3) Project MTM2011-22417, Ministerio de Ciencia e Innovacion, Spain (V. Montesinos).Zhang, ZH.; Montesinos Santalucia, V.; Liu, CY.; Gong, WZ. (2015). Geometric properties and continuity of the pre-duality mapping in Banach space. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 109(2):407-416. https://doi.org/10.1007/s13398-014-0190-6S4074161092Bandyopadhyay, P., Huang, D., Lin, B.L., Troyanski, S.L.: Some generalizations of local uniform rotundity. J. Math. Anal. Appl. 252, 906–916 (2000)Bandyopadhyay, P., Li, Y., Lin, B., Narayana, D.: Proximinality in Banach spaces. J. Math. Anal. Appl. 341, 309–317 (2008)Diestel, J.: Geometry of Banach Spaces. Selected Topics, LNM, vol. 485. Springer, Berlin (1975)Fabian, M., Habala, P., Hájek, P., Montesinos, V., Zizler, V.: Banach Space Theory. The Basis for Linear and Nonlinear Analysis, CMS Books in Mathematics. Springer, Berlin (2011)Giles, J.R., Gregory, D.A., Sims, B.: Geometrical implications of upper semi-continuity of the duality mapping on a Banach space. Pacific J. Math. 79(1), 99–109 (1978)Goebel, K., Sekowski, T.: The modulus of non-compact convexity. Ann. Univ. M. Curie-Sklodowska, Sect. A 38, 41–48 (1984)Guirao, A.J., Montesinos, V.: A note in approximative compactness and continuity of metric projections in Banach spaces. J. Convex Anal. 18, 397–401 (2011)Huff, R.: Banach spaces which are nearly uniformly convex. Rocky Mountain J. Math. 10(4), 743–749 (1980)Kutzarova, D., Rolewicz, S.: On nearly uniformly convex sets. Arch. Math. 57, 385–394 (1991)Kutzarova, D., Lin, B.L., Zhang, W.: Some geometrical properties of Banach spaces related to nearly uniform convexity. Contemp. Math. 144, 165–171 (1993)Kutzarova, D., Prus, S.: Operators which factor through nearly uniformly convex spaces. Boll. Un. Mat. Ital. B (7) 9, 2, 479–494 (1995)Montesinos, V.: Drop property equals reflexivity. Studia Math. 87, 93–100 (1987)Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, LNM, vol. 1364, 2nd edn. Springer, Berlin (1993)Rolewicz, S.: On drop property. Studia Math. 85, 27–37 (1986)Rolewicz, S.: On $$\Delta$$ Δ -uniform convexity and drop property. Studia Math. 87, 181–191 (1987)Wu, C.X., Li, Y.J.: Strong convexity in Banach spaces. J. Math. Wuhan Univ. 13(1), 105–108 (1993)Wang, J.H., Nan, C.X.: The continuity of subdifferential mapping. J. Math. Anal. Appl. 210, 206–214 (1997)Wang, J.H., Zhang, Z.H.: Characterization of the property (C-K). Acta Math. Sci. Ser. A Chin. Ed. 17(A)(3), 280–284 (1997)Zhang, Z.H., Liu, C.Y.: Some generalization of locally and weakly locally uniformly convex space. Nonlinear Anal. 74(12), 3896–3902 (2011)Zhang, Z.H., Liu, C.Y.: Convexity and proximinality in Banach spaces. J. Funct. Spaces Appl. 2012, 11 (2012). doi: 10.1155/2012/724120 . Article ID 724120Zhang, Z.H., Liu, C.Y.: Convexity and existence of the farthest point. Abstract Appl. Anal. 2011, 9 (2011). doi: 10.1155/2011/139597 . Article ID 139597Zhang, Z.H., Shi, Z.R.: Convexities and approximative compactness and continuity of the metric projection in Banach spaces. J. Approx. Theory 161(2), 802–812 (2009)Zhang, Z.H., Zhang, C.J.: On very rotund Banach spaces. Appl. Math. Mech. (English Ed.) 21(8), 965–970 (2000

Rapid Evaluation of the Special Measures for Quality and Challenged Provider Regimes: A Mixed-Methods Study

Background: Healthcare organisations in England rated as inadequate for leadership and one other domain enter Special Measures for Quality (SMQ) to receive support and oversight. A ‘watch list’ of challenged providers (CPs) at risk of entering SMQ also receive support. Knowledge is limited about whether the support interventions drive improvements in quality, their costs, and whether they strike the right balance between support and scrutiny. Objective: Analyse trust responses to the implementation of a) interventions for SMQ trusts and b) interventions for CP trusts to determine their impact on these organisations' capacity to achieve and sustain quality improvements. Design: Rapid research comprising five inter-related workstreams: 1. Literature review using systematic methods. 2. Analysis of policy documents and interviews at national level. 3. Eight multi-site, mixed method trust case studies. 4. Analysis of national performance and workforce indicators. 5. Economic analysis. Results: SMQ/CP were intended to be “support” programmes. SMQ/CP had an emotional impact on staff. Perceptions of NHSI interventions were mixed overall. Senior leadership teams were a key driver of change, with strong clinical input vital. Local systems have a role in improvement. Trusts focus efforts to improve across multiple domains. Internal and external factors contribute to positive performance trajectories. Nationally, only 15.8% of SMQ trusts exited within 24 months. Relative to national trends, entry into SMQ/CP corresponded to positive changes in 4-hour waits in Emergency Departments, mortality and delayed transfers of care. Trends in staff sickness and absence improved after trusts left SMQ/CP. There was some evidence that staff survey results improve. No association was found between SMQ/CP and referral to treatment times or cancer waiting times. The largest components of NHSI spending in case studies were interventions directed at 'training on cultural change' (33.6%), 'workforce quality and safety' (21.7%) and 'governance and assurance' (18.4%). Impact of SMQ on financial stability was equivocal; most trusts exiting SMQ experienced the same financial stability before and after exiting. Limitations: The rapid research design and one-year timeframe precludes longitudinal observations of trusts and local systems. The small number of indicators limited the quantitative analysis of impact. Measuring workforce effects was limited by data availability. Conclusions: Empirical evidence of positive impacts from SMQ/CP were identified, however, perceptions were mixed. Key lessons: • Time is needed to implement and embed changes. • Ways to mitigate emotional costs and stigma are needed. • Support strategies should be more trust specific. • Poor organisational performance needs to be addressed within local systems. • Senior leadership teams with stability, strong clinical input and previous SMQ experience helped enact change. • Organisation-wide quality improvement strategies and capabilities are needed. • Staff engagement and an open listening culture promote continuous learning and a quality improvement ‘mindset’, critical for sustainable improvement. • Need to consider level of sustainable funds required to improve patients’ outcomes. Future work: Evaluating recent changes to the regimes; role of local systems; longitudinal approaches. Study registration: Review protocol registered with PROSPERO (CRD: 42019131024). Funding: The National Institute for Health Research Health Services and Delivery Research programme (16/138/17 – Rapid Service Evaluation Research Team)

Linear Operator Inequality and Null Controllability with Vanishing Energy for unbounded control systems

We consider linear systems on a separable Hilbert space $H$, which are null controllable at some time $T_0>0$ under the action of a point or boundary control. Parabolic and hyperbolic control systems usually studied in applications are special cases. To every initial state $y_0 \in H$ we associate the minimal "energy" needed to transfer $y_0$ to $0$ in a time $T \ge T_0$ ("energy" of a control being the square of its $L^2$ norm). We give both necessary and sufficient conditions under which the minimal energy converges to $0$ for $T\to+\infty$. This extends to boundary control systems the concept of null controllability with vanishing energy introduced by Priola and Zabczyk (Siam J. Control Optim. 42 (2003)) for distributed systems. The proofs in Priola-Zabczyk paper depend on properties of the associated Riccati equation, which are not available in the present, general setting. Here we base our results on new properties of the quadratic regulator problem with stability and the Linear Operator Inequality.Comment: In this version we have also added a section on examples and applications of our main results. This version is similar to the one which will be published on "SIAM Journal on Control and Optimization" (SIAM

Chaotic differential operators

We give sufficient conditions for chaos of (differential) operators on Hilbert spaces of entire functions. To this aim we establish conditions on the coefficients of a polynomial P(z) such that P(B) is chaotic on the space lp, where B is the backward shift operator. © 2011 Springer-Verlag.This work was partially supported by the MEC and FEDER Projects MTM2007-64222, MTM2010-14909, and by GVA Project GV/2010/091, and by UPV Project PAID-06-09-2932. The authors would like to thank A. Peris for helpful comments and ideas that produced a great improvement of the paper's presentation. We also thank the referees for their helpful comments and for reporting to us a gap in Theorem 1.Conejero Casares, JA.; Martínez Jiménez, F. (2011). Chaotic differential operators. Revista- Real Academia de Ciencias Exactas Fisicas Y Naturales Serie a Matematicas. 105(2):423-431. https://doi.org/10.1007/s13398-011-0026-6S4234311052Bayart, F., Matheron, É.: Dynamics of Linear Operators, Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bermúdez T., Miller V.G.: On operators T such that f(T) is hypercyclic. Integr. Equ. Oper. Theory 37(3), 332–340 (2000)Bonet J., Martínez-Giménez F., Peris A.: Linear chaos on Fréchet spaces. Int. J. Bifur. Chaos Appl. Sci. Eng. 13(7), 1649–1655 (2003)Chan K.C., Shapiro J.H.: The cyclic behavior of translation operators on Hilbert spaces of entire functions. Indiana Univ. Math. J. 40(4), 1421–1449 (1991)Conejero J.A., Müller V.: On the universality of multipliers on $${\mathcal{H}({\mathbb {C}})}$$ . J. Approx. Theory. 162(5), 1025–1032 (2010)deLaubenfels R., Emamirad H.: Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dyn. Syst. 21(5), 1411–1427 (2001)Devaney, R.L.: An introduction to chaotic dynamical systems, 2nd edn. In: Addison-Wesley Studies in Nonlinearity. Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)Godefroy G., Shapiro J.H.: Operators with dense, invariant, cyclic vector manifolds. J. Funct. Anal. 98(2), 229–269 (1991)Grosse-Erdmann K.-G.: Hypercyclic and chaotic weighted shifts. Stud. Math. 139(1), 47–68 (2000)Grosse-Erdmann, K.-G., Peris, A.,: Linear chaos. Universitext, Springer, New York (to appear, 2011)Herzog G., Schmoeger C.: On operators T such that f(T) is hypercyclic. Stud. Math. 108(3), 209–216 (1994)Kahane, J.-P.: Some random series of functions, 2nd edn. In: Cambridge Studies in Advanced Mathematics, vol. 5. Cambridge University Press, Cambridge (1985)Martínez-Giménez F., Peris A.: Chaos for backward shift operators. Int. J. Bifur. Chaos Appl. Sci. Eng. 12(8), 1703–1715 (2002)Martínez-Giménez F.: Chaos for power series of backward shift operators. Proc. Am. Math. Soc. 135, 1741–1752 (2007)Müller V.: On the Salas theorem and hypercyclicity of f(T). Integr. Equ. Oper. Theory 67(3), 439–448 (2010)Protopopescu V., Azmy Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2(1), 79–90 (1992)Rolewicz S.: On orbits of elements. Stud. Math. 32, 17–22 (1969)Salas H.N.: Hypercyclic weighted shifts. Trans. Am. Math. Soc. 347(3), 93–1004 (1995)Shapiro, J.H.: Simple connectivity and linear chaos. Rend. Circ. Mat. Palermo. (2) Suppl. 56, 27–48 (1998

Rapid evaluation of the Special Measures for Quality and challenged provider regimes: a mixed-methods study

Background: Health-care organisations in England that are rated as inadequate for leadership and one other domain enter the Special Measures for Quality regime to receive support and oversight. A ‘watch list’ of challenged providers that are at risk of entering Special Measures for Quality also receive support. Knowledge is limited about whether or not the support interventions drive improvements in quality, the costs of the support interventions and whether or not the support interventions strike the right balance between support and scrutiny. Objective: To analyse the responses of trusts to the implementation of (1) interventions for Special Measures for Quality trusts and (2) interventions for challenged provider trusts to determine their impact on these organisations’ capacity to achieve and sustain quality improvements. Design: This was rapid research comprising five interrelated workstreams: (1) a literature review using systematic methods; (2) an analysis of policy documents and interviews at the national level; (3) eight multisite, mixed-methods trust case studies; (4) an analysis of national performance and workforce indicators; and (5) an economic analysis. Results: The Special Measures for Quality/challenged provider regimes were intended to be ‘support’ programmes. Special Measures for Quality/challenged provider regimes had an emotional impact on staff. Perceptions of NHS Improvement interventions were mixed overall. Senior leadership teams were a key driver of change, with strong clinical input being vital. Local systems have a role in improvement. Trusts focus efforts to improve across multiple domains. Internal and external factors contribute to positive performance trajectories. Nationally, only 15.8% of Special Measures for Quality trusts exited the regime in 24 months. Entry into Special Measures for Quality/challenged provider regimes resulted in changes in quality indicators (such the number of patients waiting in emergency departments for more than 4 hours, mortality and the number of delayed transfers of care) that were more positive than national trends. The trends in staff sickness and absence improved after trusts left Special Measures for Quality/challenged provider regimes. There was some evidence that staff survey results improved. No association was found between Special Measures for Quality/challenged provider regimes and referral to treatment times or cancer treatment waiting times. NHS Improvement spending in case study trusts was mostly directed at interventions addressing ‘training on cultural change’ (33.6%), ‘workforce quality and safety’ (21.7%) and ‘governance and assurance’ (18.4%). The impact of Special Measures for Quality on financial stability was equivocal; most trusts exiting Special Measures for Quality experienced the same financial stability before and after exiting. Limitations: The rapid research design and 1-year time frame precludes longitudinal observations of trusts and local systems. The small number of indicators limited the quantitative analysis of impact. Measurement of workforce effects was limited by data availability. Conclusions: Empirical evidence of positive impacts of Special Measures for Quality/challenged provider regimes were identified; however, perceptions were mixed. Key lessons were that (1) time is needed to implement and embed changes; (2) ways to mitigate emotional costs and stigma are needed; (3) support strategies should be more trust specific; (4) poor organisational performance needs to be addressed within local systems; (5) senior leadership teams with stability, strong clinical input and previous Special Measures for Quality experience helped to enact change; (6) organisation-wide quality improvement strategies and capabilities are needed; (7) staff engagement and an open-listening culture promote continuous learning and a quality improvement ‘mindset’, which is critical for sustainable improvement; and (8) consideration of the level of sustainable funds required to improve patients’ outcomes is needed. Future work: Future work could include evaluating recent changes to the regimes, the role of local systems and longitudinal approaches. Study registration: The review protocol is registered with PROSPERO (CRD42019131024). Funding: This project was funded by the National Institute for Health and Care Research (NIHR) Health and Social Care Delivery Research programme and will be published in full in Health and Social Care Delivery Research; Vol. 11, No. 19. See the NIHR Journals Library website for further project information

Local Hardy Spaces of Musielak-Orlicz Type and Their Applications

Let \phi: \mathbb{R}^n\times[0,\fz)\rightarrow[0,\fz) be a function such that $\phi(x,\cdot)$ is an Orlicz function and $\phi(\cdot,t)\in A^{\mathop\mathrm{loc}}_{\infty}(\mathbb{R}^n)$ (the class of local weights introduced by V. S. Rychkov). In this paper, the authors introduce a local Hardy space $h_{\phi}(\mathbb{R}^n)$ of Musielak-Orlicz type by the local grand maximal function, and a local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}_{\phi}(\mathbb{R}^n)$ which is further proved to be the dual space of $h_{\phi}(\mathbb{R}^n)$. As an application, the authors prove that the class of pointwise multipliers for the local $\mathop\mathrm{BMO}$-type space $\mathop\mathrm{bmo}^{\phi}(\mathbb{R}^n)$, characterized by E. Nakai and K. Yabuta, is just the dual of L^1(\rn)+h_{\Phi_0}(\mathbb{R}^n), where $\phi$ is an increasing function on $(0,\infty)$ satisfying some additional growth conditions and $\Phi_0$ a Musielak-Orlicz function induced by $\phi$. Characterizations of $h_{\phi}(\mathbb{R}^n)$, including the atoms, the local vertical and the local nontangential maximal functions, are presented. Using the atomic characterization, the authors prove the existence of finite atomic decompositions achieving the norm in some dense subspaces of $h_{\phi}(\mathbb{R}^n)$, from which, the authors further deduce some criterions for the boundedness on $h_{\phi}(\mathbb{R}^n)$ of some sublinear operators. Finally, the authors show that the local Riesz transforms and some pseudo-differential operators are bounded on $h_{\phi}(\mathbb{R}^n)$.Comment: Sci. China Math. (to appear