On the continuous Cesàro operator in certain function spaces

“The final publication is available at Springer via http://dx.doi.org/10.1007/s11117-014-0321-5"Various properties of the (continuous) Cesàro operator C, acting on Banach and Fréchet spaces of continuous functions and L p-spaces, are investigated. For instance, the spectrum and point spectrum of C are completely determined and a study of certain dynamics of C is undertaken (eg. hyper- and supercyclicity, chaotic behaviour). In addition, the mean (and uniform mean) ergodic nature of C acting in the various spaces is identified.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. (2015). On the continuous Cesàro operator in certain function spaces. Positivity. 19:659-679. https://doi.org/10.1007/s11117-014-0321-5S65967919Albanese, A.A.: Primary products of Banach spaces. Arch. Math. 66, 397–405 (1996)Albanese, A.A.: On subspaces of the spaces $$L^p_{\rm loc}$$ L loc p and of their strong duals. Math. Nachr. 197, 5–18 (1999)Albanese, A.A., Moscatelli, V.B.: Complemented subspaces of sums and products of copies of $$L^1 [0,1]$$ L 1 [ 0 , 1 ] . Rev. Mat. Univ. Complut. Madr. 9, 275–287 (1996)Albanese, 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.: On mean ergodic operators. In: Curbera, G.P. (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.: $$C_0$$ C 0 -semigroups and mean ergodic operators in a class of Fréchet spaces. J. Math. Anal. Appl. 365, 142–157 (2010)Albanese, A.A., Bonet, J., Ricker, W.J.: Convergence of arithmetic means of operators in Fréchet spaces. J. Math. Anal. Appl. 401, 160–173 (2013)Bayart, F., Matheron, E.: Dynamics of linear operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Bellenot, S.F., Dubinsky, E.: Fréchet spaces with nuclear Köthe quotients. Trans. Am. Math. Soc. 273, 579–594 (1982)Bonet, J., Frerick, L., Peris, A., Wengenroth, J.: Transitive and hypercyclic operators on locally convex spaces. Bull. Lond. Math. Soc. 37, 254–264 (2005)Boyd, D.W.: The spectrum of the Cesàro operator. Acta Sci. Math. (Szeged) 29, 31–34 (1968)Brown, A., Halmos, P.R., Shields, A.L.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125–137 (1965)Dierolf, S., Zarnadze, D.N.: A note on strictly regular Fréchet spaces. Arch. Math. 42, 549–556 (1984)Dunford, N., Schwartz, J.T.: Linear Operators I: General Theory (2nd Printing). Wiley-Interscience, New York (1964)Galaz Fontes, F., Solís, F.J.: Iterating the Cesàro operators. Proc. Am. Math. Soc. 136, 2147–2153 (2008)Galaz Fontes, F., Ruiz-Aguilar, R.W.: Grados de ciclicidad de los operadores de Cesàro–Hardy. Misc. Mat. 57, 103–117 (2013)González, M., León-Saavedra, F.: Cyclic behaviour of the Cesàro operator on $$L_2(0,+\infty )$$ L 2 ( 0 , + ∞ ) . Proc. Am. Math. Soc. 137, 2049–2055 (2009)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos. In: Universitext. Springer, London (2011)Hardy, G.H., Littlewood, J.E., Pólya, G.: Inequalities. In: Reprint of the 1952 Edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)Krengel, U.: Ergodic theorems. In: De Gruyter Studies in Mathematics, vol. 6. Walter de Gruyter Co., Berlin (1985)Leibowitz, G.M.: Spectra of finite range Cesàro operators. Acta Sci. Math. (Szeged) 35, 27–28 (1973)Leibowitz, G.M.: The Cesàro operators and their generalizations: examples in infinite-dimensional linear analysis. Am. Math. Mon. 80, 654–661 (1973)León-Saavedra, F., Piqueras-Lerena, A., Seoane-Sepúlveda, J.B.: Orbits of Cesàro type operators. Math. Nachr. 282, 764–773 (2009)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)Meise, R., Vogt, D.: Introduction to functional analysis. In: Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press; Oxford University Press, New York (1997)Metafune, G., Moscatelli, V.B.: Quojections and prequojections. In: Terzioğlu, T. (ed.) Advances in the Theory of Fréchet spaces. NATO ASI Series, vol. 287, pp. 235–254. Kluwer Academic Publishers, Dordrecht (1989)Moscatelli, V.B.: Fréchet spaces without norms and without bases. Bull. Lond. Math. Soc. 12, 63–66 (1980)Piszczek, K.: Quasi-reflexive Fréchet spaces and mean ergodicity. J. Math. Anal. Appl. 361, 224–233 (2010)Piszczek, K.: Barrelled spaces and mean ergodicity. Rev R. Acad. Cienc. Exactas Fis. Nat. Ser. A Math. RACSAM 104, 5–11 (2010)Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980

Galanin Receptor 1 Deletion Exacerbates Hippocampal Neuronal Loss after Systemic Kainate Administration in Mice

Galanin is a neuropeptide with a wide distribution in the central and peripheral nervous systems and whose physiological effects are mediated through three G protein-coupled receptor subtypes, GalR1, GalR2, and GalR3. Several lines of evidence indicate that galanin, as well as activation of the GalR1 receptor, is a potent and effective modulator of neuronal excitability in the hippocampus.In order to test more formally the potential influence of GalR1 on seizure-induced excitotoxic cell death, we conducted functional complementation tests in which transgenic mice that exhibit decreased expression of the GalR1 candidate mRNA underwent kainate-induced status epilepticus to determine if the quantitative trait of susceptibility to seizure-induced cell death is determined by the activity of GalR1. In the present study, we report that reduction of GalR1 mRNA via null mutation or injection of the GalR1 antagonist, galantide, prior to kainate-induced status epilepticus induces hippocampal damage in a mouse strain known to be highly resistant to kainate-induced neuronal injury. Wild-type and GalR1 knockout mice were subjected to systemic kainate administration. Seven days later, Nissl and NeuN immune- staining demonstrated that hippocampal cell death was significantly increased in GalR1 knockout strains and in animals injected with the GalR1 antagonist. Compared to GalR1-expressing mice, GalR1-deficient mice had significantly larger hippocampal lesions after status epilepticus.Our results suggest that a reduction of GalR1 expression in the C57BL/6J mouse strain renders them susceptible to excitotoxic injury following systemic kainate administration. From these results, GalR1 protein emerges as a new molecular target that may have a potential therapeutic value in modulating seizure-induced cell death

Physiogenomic analysis of weight loss induced by dietary carbohydrate restriction

BACKGROUND: Diets that restrict carbohydrate (CHO) have proven to be a successful dietary treatment of obesity for many people, but the degree of weight loss varies across individuals. The extent to which genetic factors associate with the magnitude of weight loss induced by CHO restriction is unknown. We examined associations among polymorphisms in candidate genes and weight loss in order to understand the physiological factors influencing body weight responses to CHO restriction. METHODS: We screened for genetic associations with weight loss in 86 healthy adults who were instructed to restrict CHO to a level that induced a small level of ketosis (CHO ~10% of total energy). A total of 27 single nucleotide polymorphisms (SNPs) were selected from 15 candidate genes involved in fat digestion/metabolism, intracellular glucose metabolism, lipoprotein remodeling, and appetite regulation. Multiple linear regression was used to rank the SNPs according to probability of association, and the most significant associations were analyzed in greater detail. RESULTS: Mean weight loss was 6.4 kg. SNPs in the gastric lipase (LIPF), hepatic glycogen synthase (GYS2), cholesteryl ester transfer protein (CETP) and galanin (GAL) genes were significantly associated with weight loss. CONCLUSION: A strong association between weight loss induced by dietary CHO restriction and variability in genes regulating fat digestion, hepatic glucose metabolism, intravascular lipoprotein remodeling, and appetite were detected. These discoveries could provide clues to important physiologic adaptations underlying the body mass response to CHO restriction

The Insulin-Mediated Modulation of Visually Evoked Magnetic Fields Is Reduced in Obese Subjects

BACKGROUND: Insulin is an anorexigenic hormone that contributes to the termination of food intake in the postprandial state. An alteration in insulin action in the brain, named "cerebral insulin resistance", is responsible for overeating and the development of obesity. METHODOLOGY/PRINCIPAL FINDINGS: To analyze the direct effect of insulin on food-related neuronal activity we tested 10 lean and 10 obese subjects. We conducted a magnetencephalography study during a visual working memory task in both the basal state and after applying insulin or placebo spray intranasally to bypass the blood brain barrier. Food and non-food pictures were presented and subjects had to determine whether or not two consecutive pictures belonged to the same category. Intranasal insulin displayed no effect on blood glucose, insulin or C-peptide concentrations in the periphery; however, it led to an increase in the components of evoked fields related to identification and categorization of pictures (at around 170 ms post stimuli in the visual ventral stream) in lean subjects when food pictures were presented. In contrast, insulin did not modulate food-related brain activity in obese subjects. CONCLUSIONS/SIGNIFICANCE: We demonstrated that intranasal insulin increases the cerebral processing of food pictures in lean whereas this was absent in obese subjects. This study further substantiates the presence of a "cerebral insulin resistance" in obese subjects and might be relevant in the pathogenesis of obesity

Fluoxetine during Development Reverses the Effects of Prenatal Stress on Depressive-Like Behavior and Hippocampal Neurogenesis in Adolescence

Depression during pregnancy and the postpartum period is a growing health problem, which affects up to 20% of women. Currently, selective serotonin reuptake inhibitor (SSRIs) medications are commonly used for treatment of maternal depression. Unfortunately, there is very little research on the long-term effect of maternal depression and perinatal SSRI exposure on offspring development. Therefore, the aim of this study was to determine the role of exposure to fluoxetine during development on affective-like behaviors and hippocampal neurogenesis in adolescent offspring in a rodent model of maternal depression. To do this, gestationally stressed and non-stressed Sprague-Dawley rat dams were treated with either fluoxetine (5 mg/kg/day) or vehicle beginning on postnatal day 1 (P1). Adolescent male and female offspring were divided into 4 groups: 1) prenatal stress+fluoxetine exposure, 2) prenatal stress+vehicle, 3) fluoxetine exposure alone, and 4) vehicle alone. Adolescent offspring were assessed for anxiety-like behavior using the Open Field Test and depressive-like behavior using the Forced Swim Test. Brains were analyzed for endogenous markers of hippocampal neurogenesis via immunohistochemistry. Results demonstrate that maternal fluoxetine exposure reverses the reduction in immobility evident in prenatally stressed adolescent offspring. In addition, maternal fluoxetine exposure reverses the decrease in hippocampal cell proliferation and neurogenesis in maternally stressed adolescent offspring. This research provides important evidence on the long-term effect of fluoxetine exposure during development in a model of maternal adversity

Serotonin transporter binding of [123I]ADAM in bulimic women, their healthy twin sisters, and healthy women: a SPET study

<p>Abstract</p> <p>Background</p> <p>Bulimia Nervosa (BN) is believed to be caused by an interaction of genetic and environmental factors. Previous studies support the existence of a bulimia-related endophenotype as well as disturbances in serotonin (5-HT) transmission. We studied serotonin transporter (SERT) binding in BN, and to investigate the possibility of a SERT-related endophenotype for BN, did this in a sample of female twins. We hypothesized clearly reduced SERT binding in BN women as opposed to healthy women, and intermediate SERT binding in unaffected co-twins.</p> <p>Methods</p> <p>We studied 13 female twins with BN (9 with purging and 4 with non-purging BN) and 25 healthy women, including 6 healthy twin sisters of BN patients and 19 women from 10 healthy twin pairs. [¹²³I]ADAM, a selective SERT radioligand for single photon emission tomography (SPET) imaging, was used to assess SERT availability in the midbrain and the thalamus.</p> <p>Results</p> <p>No differences in SERT binding were evident when comparing the BN women, their unaffected co-twins and the healthy controls (p = 0.14). The healthy sisters of the BN patients and the healthy control women had similar SERT binding in both brain regions. In a <it>post hoc </it>subgroup analysis, the purging bulimics had higher SERT binding than the healthy women in the midbrain (p = 0.03), but not in the thalamus.</p> <p>Conclusion</p> <p>Our finding of increased SERT binding in the midbrain in the purging BN women raises the possibility that this subgroup of bulimics might differ in serotonergic function from the non-purging ones. The similarity of the unaffected co-twins and the healthy controls doesn't support our initial assumption of a SERT-related endophenotype for BN. Due to the small sample size, our results need to be interpreted with caution and verified in a larger sample.</p

Polymorphisms of the TUB Gene Are Associated with Body Composition and Eating Behavior in Middle-Aged Women

BACKGROUND: The TUB gene, encoding an evolutionary conserved protein, is highly expressed in the hypothalamus and might act as a transcription factor. Mutations in TUB cause late-onset obesity, insulin-resistance and neurosensory deficits in mice. An association of common variants in the TUB gene with body weight in humans has been reported. METHODS/FINDINGS: The aim was to investigate the relationship of single nucleotide polymorphisms (SNPs) of the TUB gene (rs2272382, rs2272383 and rs1528133) with both anthropometry and self-reported macronutrient intake from a validated food frequency questionnaire. These associations were studied in a population-based, cross-sectional study of 1680 middle-aged Dutch women, using linear regression analysis. The minor allele C of the rs1528133 SNP was significantly associated with increased weight (+1.88 kg, P = 0.022) and BMI (+0.56 units, P = 0.05). Compared with non-carriers, both AG heterozygotes and AA homozygotes of the rs2272382 SNP derived less energy from fat (AG: -0.55+/-0.28%, P = 0.05, AA: -0.95+/-0.48%, P = 0.047). However, both genotypes were associated with an increased energy intake from carbohydrates (0.69+/-0.33%, P = 0.04 and 1.68+/-0.56%, P = 0.003, respectively), mainly because of a higher consumption of mono- and disaccharides. Both these SNPs, rs2272382 and rs1528133, were also associated with a higher glycemic load in the diet. The glycemic load was higher among those with AG and AA genotypes for the variant rs2272382 than among the wild types (+1.49 (95% CI: -0.27-3.24) and +3.89 (95% CI: 0.94-6.85) units, respectively). Carriers of the minor allele C of rs1528133 were associated with an increased glycemic load of 1.85 units compared with non-carriers. CONCLUSIONS: Genetic variation of the TUB gene was associated with both body composition and macronutrient intake, suggesting that TUB might influence eating behavior

Admixture Mapping of 15,280 African Americans Identifies Obesity Susceptibility Loci on Chromosomes 5 and X

The prevalence of obesity (body mass index (BMI) ≥30 kg/m2) is higher in African Americans than in European Americans, even after adjustment for socioeconomic factors, suggesting that genetic factors may explain some of the difference. To identify genetic loci influencing BMI, we carried out a pooled analysis of genome-wide admixture mapping scans in 15,280 African Americans from 14 epidemiologic studies. Samples were genotyped at a median of 1,411 ancestry-informative markers. After adjusting for age, sex, and study, BMI was analyzed both as a dichotomized (top 20% versus bottom 20%) and a continuous trait. We found that a higher percentage of European ancestry was significantly correlated with lower BMI (ρ = −0.042, P = 1.6×10−7). In the dichotomized analysis, we detected two loci on chromosome X as associated with increased African ancestry: the first at Xq25 (locus-specific LOD = 5.94; genome-wide score = 3.22; case-control Z = −3.94); and the second at Xq13.1 (locus-specific LOD = 2.22; case-control Z = −4.62). Quantitative analysis identified a third locus at 5q13.3 where higher BMI was highly significantly associated with greater European ancestry (locus-specific LOD = 6.27; genome-wide score = 3.46). Further mapping studies with dense sets of markers will be necessary to identify the alleles in these regions of chromosomes X and 5 that may be associated with variation in BMI