Characterizing Fréchet-Schwartz spaces via power bounded operators

[EN] We characterize Köthe echelon spaces (and, more generally, those Fréchet spaces with an unconditional basis) which are Schwartz, in terms of the convergence of the Cesàro means of power bounded operators defined on them. This complements similar known characterizations of reflexive and of Fréchet–Montel spaces with a basis. Every strongly convergent sequence of continuous linear operators on a FréchetSchwartz space does so in a special way. We single out this type of “rapid convergence” for a sequence of operators and study its relationship to the structure of the underlying space. Its relevance for Schauder decompositions and the connection to mean ergodic operators on Fréchet–Schwartz spaces is also investigated.The research of the first two authors was partially supported by the projects MTM2010-15200 and GVA Prometeo II/2013/013 (Spain). The second author gratefully acknowledges the support of the Alexander von Humboldt Foundation.Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2014). Characterizing Fréchet-Schwartz spaces via power bounded operators. Studia Mathematica. 224(1):25-45. https://doi.org/10.4064/sm224-1-22545224

Mean ergodic semigroups of operators

We present criteria for determining mean ergodicity of C 0-semigroups of linear operators in a sequentially complete, locally convex Hausdorff space X. A characterization of reflexivity of certain spaces X with a basis via mean ergodicity of equicontinuous C 0-semigroups acting in X is also presented. Special results become available in Grothendieck spaces with the Dunford-Pettis property. © 2011 Springer-Verlag.Research partially supported by MICINN and FEDER Project MTM2010-15200 and GV Project Prometeo/2008/101.Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2012). Mean ergodic semigroups of operators. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas. 106(2):299-319. https://doi.org/10.1007/s13398-011-0054-2S2993191062Albanese A.A., Bonet J., Ricker W.J.: Mean ergodic operators in Fréchet spaces. Ann. Acad. Sci. Fenn. Math. 34, 401–436 (2009)Albanese A.A., Bonet J., Ricker W.J.: Grothendieck spaces with the Dunford–Pettis property. Positivity 14, 145–164 (2010)Albanese, A.A., Bonet, J., Ricker, W.J.: On mean ergodic operators. In: Curbera, G.P. et al. (eds.) Vector Measures, Integration and Related Topics. Operator Theory: Advances and Applications, vol. 201, pp. 1–20. Birkhäuser, Basel (2010)Albanese A.A., Bonet J., Ricker W.J.: C 0-semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)Bonet J., Ricker W.J.: Schauder decompositions and the Grothendieck and the Dunford–Pettis properties in Köthe echelon spaces of infinite order. Positivity 11, 77–93 (2007)Bonet J., Ricker W.J.: Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions. Arch. Math. 92, 428–437 (2009)Bonet J., de Pagter B., Ricker W.J.: Mean ergodic operators and reflexive Fréchet lattices. Proc. R. Soc. Edinb. Sect. A 141, 897–920 (2011)Dunford N., Schwartz J.T.: Linear Operators I: General Theory, 2nd edn. Wiley, New York (1964)Eberlein W.F.: Abstract ergodic theorems and weak almost periodic functions. Trans. Am. Math. Soc. 67, 217–240 (1949)Edwards R.E.: Functional Analysis. Reinhart and Winston, New York (1965)Engel K.-J., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (1999)Floret, K.: Weakly compact sets. In: LNM, vol. 801. Springer, Berlin (1980)Fonf V.P., Lin M., Wojtaszczyk P.: Ergodic characterizations of reflexivity in Banach spaces. J. Funct. Anal. 187, 146–162 (2001)Hille, E., Phillips, R.S.: Functional Analysis and Semigroups, 4th edn. American Math. Soc., Providence (1981, revised)Jarchow H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981)Kelley J.L.: General Topology, Rev. Edn, D. van Nostrand Co., Princeton–New York (1961)Komatsu H.: Semi-groups of operators in locally convex spaces. J. Math. Soc. Japan 16, 230–262 (1964)Komura T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)Köthe, G.: Topological Vector Spaces I, 2nd edn. Springer, Berlin (1983, revised)Köthe G.: Topological Vector Spaces II. Springer, Berlin (1979)Krengel U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)Lotz, H.P. (1984) Tauberian theorems for operators on L ∞ and similar spaces. In: Bierstedt, K.D., Fuchssteiner, B. (eds.) Functional Analyis: Surveys and Recent Results III. North Holland, Amsterdam, pp. 117–133Lotz H.P.: Uniform convergence of operators on L ∞ and similar spaces. Math. Z. 190, 207–220 (1985)Meise R., Vogt D.: Introduction to Functional Analysis. Clarendon Press, Oxford (1997)Miyadera I.: Semigroups of operators in Fréchet spaces and applications to partial differential operators. Tôhoku Math. J. 11, 162–183 (1959)Mugnolo D.: A semigroup analogue of the Fonf–Lin–Wojtaszczyk characterization of reflexive Banach spaces with a basis. Studia Math. 164, 243–251 (2004)Piszczek K.: Quasi-reflexive Fréchet spaces and mean ergodicity. J. Math. Anal. Appl. 361, 224–233 (2010)Piszczek K.: Barrelled spaces and mean ergodicity. RACSAM 104, 5–11 (2010)Rudin W.: Functional Analysis. McGraw-Hill, New York (1973)Sato R.: On a mean ergodic theorem. Proc. Am. Math. Soc. 83, 563–564 (1981)Schaefer H.H.: Banach Lattices and Positive Operators. Springer, Berlin (1974)Shaw S.-Y.: Ergodic projections of continuous and discrete semigroups. Proc. Am. Math. Soc. 78, 69–76 (1980)Yosida K.: Functional Analysis. Springer, Berlin (1980

Duration of Posttraumatic Amnesia Predicts Neuropsychological and Global Outcome in Complicated Mild Traumatic Brain Injury.

OBJECTIVES: Examine the effects of posttraumatic amnesia (PTA) duration on neuropsychological and global recovery from 1 to 6 months after complicated mild traumatic brain injury (cmTBI). PARTICIPANTS: A total of 330 persons with cmTBI defined as Glasgow Coma Scale score of 13 to 15 in emergency department, with well-defined abnormalities on neuroimaging. METHODS: Enrollment within 24 hours of injury with follow-up at 1, 3, and 6 months. MEASURES: Glasgow Outcome Scale-Extended, California Verbal Learning Test II, and Controlled Oral Word Association Test. Duration of PTA was retrospectively measured with structured interview at 30 days postinjury. RESULTS: Despite all having a Glasgow Coma Scale Score of 13 to 15, a quarter of the sample had a PTA duration of greater than 7 days; half had PTA duration of 1 of 7 days. Both cognitive performance and Extended Glasgow Outcome Scale outcomes were strongly associated with time since injury and PTA duration, with those with PTA duration of greater than 1 week showing residual moderate disability at 6-month assessment. CONCLUSIONS: Findings reinforce importance of careful measurement of duration of PTA to refine outcome prediction and allocation of resources to those with cmTBI. Future research would benefit from standardization in computed tomographic criteria and use of severity indices beyond Glasgow Coma Scale to characterize cmTBI

Increased burden of familial-associated early-onset cancer risk among minority americans compared to non-latino whites

Background: The role of race/ethnicity in genetic predisposition of early-onset cancers can be estimated by comparing family-based cancer concordance rates among ethnic groups. Methods: We used linked California health registries to evaluate the relative cancer risks for first-degree relatives of patients diagnosed between ages 0 and 26, and the relative risks of developing distinct second primary malignancies (SPMs). From 1989 to 2015, we identified 29,631 cancer patients and 62,863 healthy family members. We calculated the standardized incident ratios (SIRs) of early-onset primary cancers diagnosed in proband siblings and mothers, as well as SPMs detected among early-onset patients. Analyses were stratified by self-identified race/ethnicity. Results: Given probands with cancer, there were increased relative risks of any cancer for siblings and mothers (SIR = 3.32; 95% confidence interval [CI]: 2.85–3.85) and of SPMs (SIR = 7.27; 95% CI: 6.56–8.03). Given a proband with solid cancer, both Latinos (SIR = 4.98; 95% CI: 3.82–6.39) and non-Latino Blacks (SIR = 7.35; 95% CI: 3.36–13.95) exhibited significantly higher relative risk of any cancer in siblings and mothers when compared to non-Latino White subjects (SIR = 3.02; 95% CI: 2.12–4.16). For hematologic cancers, higher familial risk was evident for Asian/Pacific Islanders (SIR = 7.56; 95% CI: 3.26–14.90) compared to non-Latino whites (SIR = 2.69; 95% CI: 1.62–4.20). Conclusions: The data support a need for increased attention to the genetics of early-onset cancer predisposition and environmental factors in race/ethnic minority families in the United States.</p

Three red suns in the sky: A transiting, terrestrial planet in a triple M-dwarf system at 6.9 pc

We present the discovery from Transiting Exoplanet Survey Satellite (TESS) data of LTT 1445Ab. At a distance of 6.9 pc, it is the second nearest transiting exoplanet system found to date, and the closest one known for which the primary is an M dwarf. The host stellar system consists of three mid-to-late M dwarfs in a hierarchical configuration, which are blended in one TESS pixel. We use MEarth data and results from the Science Processing Operations Center data validation report to determine that the planet transits the primary star in the system. The planet has a radius of ${1.38}_{-0.12}^{+0.13}$ ${R}_{\oplus }$, an orbital period of ${5.35882}_{-0.00031}^{+0.00030}$ days, and an equilibrium temperature of ${433}_{-27}^{+28}$ K. With radial velocities from the High Accuracy Radial Velocity Planet Searcher, we place a 3σ upper mass limit of 8.4 ${M}_{\oplus }$ on the planet. LTT 1445Ab provides one of the best opportunities to date for the spectroscopic study of the atmosphere of a terrestrial world. We also present a detailed characterization of the host stellar system. We use high-resolution spectroscopy and imaging to rule out the presence of any other close stellar or brown dwarf companions. Nineteen years of photometric monitoring of A and BC indicate a moderate amount of variability, in agreement with that observed in the TESS light-curve data. We derive a preliminary astrometric orbit for the BC pair that reveals an edge-on and eccentric configuration. The presence of a transiting planet in this system hints that the entire system may be co-planar, implying that the system may have formed from the early fragmentation of an individual protostellar core.Accepted manuscrip

TESS Discovery of an ultra-short-period planet around the nearby M dwarf LHS 3844

Data from the newly-commissioned \textit{Transiting Exoplanet Survey Satellite} (TESS) has revealed a "hot Earth" around LHS 3844, an M dwarf located 15 pc away. The planet has a radius of $1.32\pm 0.02$ $R_\oplus$ and orbits the star every 11 hours. Although the existence of an atmosphere around such a strongly irradiated planet is questionable, the star is bright enough ($I=11.9$, $K=9.1$) for this possibility to be investigated with transit and occultation spectroscopy. The star's brightness and the planet's short period will also facilitate the measurement of the planet's mass through Doppler spectroscopy.Comment: 10 pages, 4 figures. Submitted to ApJ Letters. This letter makes use of the TESS Alert data, which is currently in a beta test phase, using data from the pipelines at the TESS Science Office and at the TESS Science Processing Operations Cente

TESS Discovery of a Transiting Super-Earth in the π\pi Mensae System

We report the detection of a transiting planet around $\pi$ Mensae (HD 39091), using data from the Transiting Exoplanet Survey Satellite (TESS). The solar-type host star is unusually bright (V=5.7) and was already known to host a Jovian planet on a highly eccentric, 5.7-year orbit. The newly discovered planet has a size of $2.04\pm 0.05$ $R_\oplus$ and an orbital period of 6.27 days. Radial-velocity data from the HARPS and AAT/UCLES archives also displays a 6.27-day periodicity, confirming the existence of the planet and leading to a mass determination of $4.82\pm 0.85$ $M_\oplus$. The star's proximity and brightness will facilitate further investigations, such as atmospheric spectroscopy, asteroseismology, the Rossiter--McLaughlin effect, astrometry, and direct imaging.Comment: Accepted for publication ApJ Letters. This letter makes use of the TESS Alert data, which is currently in a beta test phase. The discovery light curve is included in a table inside the arxiv submissio

KELT-9 b's Asymmetric TESS Transit Caused by Rapid Stellar Rotation and Spin-Orbit Misalignment

KELT-9 b is an ultra hot Jupiter transiting a rapidly rotating, oblate early-A-type star in a polar orbit. We model the effect of rapid stellar rotation on KELT-9 b's transit light curve using photometry from the Transiting Exoplanet Survey Satellite (\tess) to constrain the planet's true spin-orbit angle and to explore how KELT-9 b may be influenced by stellar gravity darkening. We constrain the host star's equatorial radius to be $1.089\pm0.017$ times as large as its polar radius and its local surface brightness to vary by $\sim38$\% between its hot poles and cooler equator. We model the stellar oblateness and surface brightness gradient and find that it causes the transit light curve to lack the usual symmetry around the time of minimum light. We take advantage of the light curve asymmetry to constrain KELT-9 b's true spin orbit angle (${87^\circ}^{+10^\circ}_{-11^\circ}$), agreeing with \citet{gaudi2017giant} that KELT-9 b is in a nearly polar orbit. We also apply a gravity darkening correction to the spectral energy distribution model from \citet{gaudi2017giant} and find that accounting for rapid rotation gives a better fit to available spectroscopy and yields a more reliable estimate for the star's polar effective temperature.Comment: Accepted for Publication in ApJ. arXiv admin note: text overlap with arXiv:1911.0502