DUALITY FOR SOME LARGE SPACES OF ANALYTIC FUNCTIONS

We characterize the duals and biduals of the $L^p$-analogues $\mathcal{N}_\alpha^p$ of the standard Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\ge-1$ and $1\le p\lt \infty$. We adopt the convention to take $\mathcal{N}_{-1}^p$ to be the classical Smirnov class $\mathcal{N}^+$ for $p=1$, and the Hardy-Orlicz space $LH^p$ $(=(\text{Log}^+H)^p)$ for $1\lt p\lt\infty$. Our results generalize and unify earlier characterizations obtained by Eoff for $\alpha=0$ and $\alpha=-1$, and by Yanigahara for the Smirnov class. Each $\mathcal{N}_\alpha^p$ is a complete metrizable topological vector space (in fact, even an algebra); it fails to be locally bounded and locally convex but admits a separating dual. Its bidual will be identified with a specific nuclear power series space of finite type; this turns out to be the ‘Fréchet envelope' of $\mathcal{N}_\alpha^p$ as well. The generating sequence of this power series space is of the form $(n^\theta)_{n\in\mathbb{N}}$ for some $0\lt\theta\lt1$. For example, the $\theta$s in the interval (\smfr12,1) correspond in a bijective fashion to the Nevanlinna classes $\mathcal{N}_\alpha$, $\alpha\gt-1$, whereas the $\theta$s in the interval (0,\smfr12) correspond bijectively to the Hardy-Orlicz spaces $LH^p$, $1\lt p\lt \infty$. By the work of Yanagihara, \theta=\smfr12 corresponds to $\mathcal{N}^+$. As in the work by Yanagihara, we derive our results from characterizations of coefficient multipliers from $\mathcal{N}_\alpha^p$ into various smaller classical spaces of analytic functions on $\Delta$. AMS 2000 Mathematics subject classification: Primary 46E10; 46A11; 47B38. Secondary 30D55; 46A45; 46E15\vskip-3p

Searches for HCl and HF in comets 103P/Hartley 2 and C/2009 P1 (Garradd) with the Herschel space observatory

HCl and HF are expected to be the main reservoirs of fluorine and chlorine wherever hydrogen is predominantly molecular. They are found to be strongly depleted in dense molecular clouds, suggesting freeze-out onto grains in such cold environments. We can then expect that HCl and HF were also the major carriers of Cl and F in the gas and icy phases of the outer solar nebula, and were incorporated into comets. We aimed to measure the HCl and HF abundances in cometary ices as they can provide insights on the halogen chemistry in the early solar nebula. We searched for the J(1-0) lines of HCl and HF at 626 and 1232 GHz, respectively, using the HIFI instrument on board the Herschel Space Observatory. HCl was searched for in comets 103P/Hartley 2 and C/2009 P1 (Garradd), whereas observations of HF were conducted in comet C/2009 P1. In addition, observations of H$_2$O and H$_2^{18}$O lines were performed in C/2009 P1 to measure the H$_2$O production rate. Three lines of CH$_3$OH were serendipitously observed in the HCl receiver setting. HCl is not detected, whereas a marginal (3.6-$\sigma$) detection of HF is obtained. The upper limits for the HCl abundance relative to water are 0.011% and 0.022%, for 103P and C/2009 P1, respectively, showing that HCl is depleted with respect to the solar Cl/O abundance by a factor more than 6$^{+6}_{-3}$ in 103P, where the error is related to the uncertainty in the chlorine solar abundance. The marginal HF detection obtained in C/2009 P1 corresponds to an HF abundance relative to water of (1.8$\pm$0.5) $\times$ 10$^{-4}$, which is approximately consistent with a solar photospheric F/O abundance. The observed depletion of HCl suggests that HCl was not the main reservoir of chlorine in the regions of the solar nebula where these comets formed. HF was possibly the main fluorine compound in the gas phase of the outer solar nebula.Comment: Accepted for publication in Astronomy & Astrophysic

Controllability on infinite-dimensional manifolds

Following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable.Comment: 19 pages, 1 figur

Improving integrability via absolute summability: a general version of Diestel s Theorem

[EN] A classical result by J. Diestel establishes that the composition of a summing operator with a (strongly measurable) Pettis integrable function gives a Bochner integrable function. In this paper we show that a much more general result is possible regarding the improvement of the integrability of vector valued functions by the summability of the operator. After proving a general result, we center our attention in the particular case given by the -absolutely continuous operators, that allows to prove a lot of special results on integration improvement for selected cases of classical Banach spaces-including C(K), and Hilbert spaces-and operators-p-summing, (q, p)-summing and p-approximable operators.D. Pellegrino acknowledges with thanks the support of CNPq Grant 401735/2013-3 (Brazil). P. Rueda acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Spain) MTM2011-22417. E.A. Sanchez Perez acknowledges with thanks the support of the Ministerio de Economia y Competitividad (Spain) MTM2012-36740-C02-02.Pellegrino, D.; Rueda, P.; Sánchez Pérez, EA. (2016). Improving integrability via absolute summability: a general version of Diestel s Theorem. Positivity. 20(2):369-383. https://doi.org/10.1007/s11117-015-0361-5S369383202Botelho, G., Pellegrino, D., Rueda, P.: A unified Pietsch domination theorem. J. Math. Anal. Appl. 365(1), 269–276 (2010)Defant, A., Floret, K.: Tensor norms and operator ideals. North-Holland, Amsterdam (1992)Diestel, J.: An elementary characterization of absolutely summing operators. Math. Ann. 196, 101–105 (1972)Diestel, J., Jarchow, H., Tonge, A.: Absolutely summing operators. Cambridge University Press, Cambridge (1995)Farmer, J., Johnson, W.B.: Lipschitz p-summing operators. Proc. Amer. Math. Soc. 137, 2989–2995 (2009)Jarchow, H.: Localy convex, spaces. Teubner, Stuttgart (1981)López Molina, J.A., Sánchez Pérez, E.A.: Ideales de operadores absolutamente continuos, Ciencias Exactas, Físicas y Naturales, Madrid. Rev. Real Acad. 87, 349–378 (1993)López Molina, J.A., Sánchez Pérez, E.A.: The associated tensor norm to $$(q, p)$$ ( q , p ) -absolutely summing operators on $$C(K)$$ C ( K ) -spaces. Czec. Math. J. 47(4), 627–631 (1997)López, J.A., Molina, Sánchez-Pérez, E.A.: On operator ideals related to $$(p,\sigma )$$ ( p , σ ) -absolutely continuous operator. Studia Math. 131(8), 25–40 (2000)Matter, U.: Absolute continuous operators and super-reflexivity. Math. Nachr. 130, 193–216 (1987)Pellegrino, D., Santos, J.: A general Pietsch domination theorem. J. Math. Anal. Appl. 375(1), 371–374 (2011)Pellegrino, D., Santos, J., Seoane-Sepúlveda, J.B.: Some techniques on nonlinear analysis and applications. Adv. Math. 229, 1235–1265 (2012)Pietsch, A.: Operator Ideals. Deutsch. Verlag Wiss., Berlin, 1978; North-Holland, Amsterdam-London-New York-Tokyo (1980)Pisier, G.: Factorization of operators through $$L_{p\infty }$$ L p ∞ or $$L_{p1}$$ L p 1 and noncommutative generalizations. Math. Ann. 276(1), 105–136 (1986)Rodríguez, J.: Absolutely summing operators and integration of vector-valued functions. J. Math. Anal. Appl. 316(2), 579–600 (2006

Weighted composition operators on Korenblum type spaces of analytic functions

[EN] We investigate the continuity, compactness and invertibility of weighted composition operators W-psi,W-phi: f -> psi(f circle phi) when they act on the classical Korenblum space A(-infinity) and other related Frechet or (LB)-spaces of analytic functions on the open unit disc which are defined as intersections or unions of weighted Banach spaces with sup-norms. Some results about the spectrum of these operators are presented in case the self-map phi has a fixed point in the unit disc. A precise description of the spectrum is obtained in this case when the operator acts on the Korenblum space.This research was partially supported by the research project MTM2016-76647-P and the grant BES-2017-081200.Gomez-Orts, E. (2020). Weighted composition operators on Korenblum type spaces of analytic functions. Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas. 114(4):1-15. https://doi.org/10.1007/s13398-020-00924-1S1151144Abramovich, Y.A., Aliprantis, C.D.: An invitation to operator theory. Graduate Studies in Mathematics. Amer. Math. Soc., 50 (2002)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces $$\ell ^{p+}$$ and $$L^{p-}$$. Glasgow Math. J. 59, 273–287 (2017)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator on Korenblum type spaces of analytic functions. Collect. Math. 69(2), 263–281 (2018)Albanese, A.A., Bonet, J., Ricker, W.J.: Operators on the Fréchet sequence spaces $$ces(p+), 1\le p\le \infty$$. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 113(2), 1533–1556 (2019)Albanese, A.A., Bonet, J., Ricker, W.J.: Linear operators on the (LB)-sequence spaces $$ces(p-), 1\le p\le \infty$$. Descriptive topology and functional analysis. II, 43–67, Springer Proc. Math. Stat., 286, Springer, Cham (2019)Arendt, W., Chalendar, I., Kumar, M., Srivastava, S.: Powers of composition operators: asymptotic behaviour on Bergman, Dirichlet and Bloch spaces. J. Austral. Math. Soc. 1–32. https://doi.org/10.1017/S1446788719000235Aron, R., Lindström, M.: Spectra of weighted composition operators on weighted Banach spaces of analytic funcions. Israel J. Math. 141, 263–276 (2004)Bierstedt, K.D., Summers, W.H.: Biduals of weighted Banach spaces of analytic functions. J. Austral. Math. Soc., Ser. A, 54(1), 70–79 (1993)Bonet, J.: A note about the spectrum of composition operators induced by a rotation. RACSAM 114, 63 (2020). https://doi.org/10.1007/s13398-020-00788-5Bonet, J., Domański, P., Lindström, M., Taskinen, J.: Composition operators between weighted Banach spaces of analytic functions. J. Austral. Math. Soc., Ser. A, 64(1), 101–118 (1998)Bourdon, P.S.: Essential angular derivatives and maximum growth of Königs eigenfunctions. J. Func. Anal. 160, 561–580 (1998)Bourdon, P.S.: Invertible weighted composition operators. Proc. Am. Math. Soc. 142(1), 289–299 (2014)Carleson, L., Gamelin, T.: Complex Dynamics. Springer, Berlin (1991)Cowen, C., MacCluer, B.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton, FL (1995)Contreras, M., Hernández-Díaz, A.G.: Weighted composition operators in weighted Banach spacs of analytic functions. J. Austral. Math. Soc., Ser. A 69, 41–60 (2000)Eklund, T., Galindo, P., Lindström, M.: Königs eigenfunction for composition operators on Bloch and $$H^\infty$$ spaces. J. Math. Anal. Appl. 445, 1300–1309 (2017)Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Grad. Texts in Math. 199. Springer, New York (2000)Jarchow, H.: Locally Convex Spaces. Teubner, Stuttgart (1981)Kamowitz, H.: Compact operators of the form $$uC_{\varphi }$$. Pac. J. Math. 80(1) (1979)Korenblum, B.: An extension of the Nevanlinna theory. Acta Math. 135, 187–219 (1975)Köthe, G.: Topological Vector Spaces II. Springer, New York Inc (1979)Lusky, W.: On the isomorphism classes of weighted spaces of harmonic and holomophic functions. Stud. Math. 75, 19–45 (2006)Meise, R., Vogt, D.: Introduction to functional analysis. Oxford Grad. Texts in Math. 2, New York, (1997)Montes-Rodríguez, A.: Weighted composition operators on weighted Banach spaces of analytic functions. J. Lond. Math. Soc. 61(3), 872–884 (2000)Queffélec, H., Queffélec, M.: Diophantine Approximation and Dirichlet series. Hindustain Book Agency, New Delhi (2013)Shapiro, J.H.: Composition Operators and Classical Function Theory. Springer, New York (1993)Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Amer. Math. Soc. 162, 287–302 (1971)Zhu, K.: Operator Theory on Function Spaces, Math. Surveys and Monographs, Amer. Math. Soc. 138 (2007

HIFI observations of warm gas in DR21: Shock versus radiative heating

The molecular gas in the DR21 massive star formation region is known to be affected by the strong UV field from the central star cluster and by a fast outflow creating a bright shock. The relative contribution of both heating mechanisms is the matter of a long debate. By better sampling the excitation ladder of various tracers we provide a quantitative distinction between the different heating mechanisms. HIFI observations of mid-J transitions of CO and HCO+ isotopes allow us to bridge the gap in excitation energies between observations from the ground, characterizing the cooler gas, and existing ISO LWS spectra, constraining the properties of the hot gas. Comparing the detailed line profiles allows to identify the physical structure of the different components. In spite of the known shock-excitation of H2 and the clearly visible strong outflow, we find that the emission of all lines up to > 2 THz can be explained by purely radiative heating of the material. However, the new Herschel/HIFI observations reveal two types of excitation conditions. We find hot and dense clumps close to the central cluster, probably dynamically affected by the outflow, and a more widespread distribution of cooler, but nevertheless dense, molecular clumps.Comment: Accepted for publication by A&

Operators on the Fréchet sequence space ces(p+), 1≤p<∞1 \leq p < \infty

[EN] The Fréchet sequence spaces ces(p+) are very different to the Fréchet sequence spaces ¿p+,1¿pp}\ell ^q$$ ℓ p + = ∩ q > p ℓ q . Math. Nachr. 147, 7–12 (1990)Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces. North Holland, Amsterdam (1987)Pitt, H.R.: A note on bilinear forms. J. Lond. Math. Soc. 11, 171–174 (1936)Ricker, W.J.: A spectral mapping theorem for scalar-type spectral operators in locally convex spaces. Integral Equ. Oper. Theory 8, 276–288 (1985)Robertson, A.P., Robertson, W.: Topological Vector Spaces. Cambridge University Press, Cambridge (1973)Waelbroeck, L.: Topological vector spaces and algebras. Lecture Notes in Mathematics, vol. 230. Springer, Berlin (1971

Cotauberian Operators on L1(0, 1) Obtained by Lifting

ABSTRACT:We show that the set Td(L1(0, 1)) of cotauberian operators acting on L1(0, 1) is not open, and T ? Td(L1(0, 1)) does not imply T** cotauberian. As a consequence, we derive that the set T(L8(0, 1)) of tauberian operators acting on L8(0, 1) is not open, and that T ? T(L8(0,1)) does not imply T** tauberian

Dynamics and spectrum of the Cesàro operator on C-infinity(R+)

[EN] The spectrum and point spectrum of the Cesaro averaging operator C acting on the Frechet space C-infinity(R+) of all C-infinity functions on the interval [0, infinity) are determined. We employ an approach via C-0-semigroup theory for linear operators. A spectral mapping theorem for the resolvent of a closed operator acting on a locally convex space is established; it constitutes a useful tool needed to establish the main result. The dynamical behaviour of C is also investigated.The research of the first two authors was partially supported by the projects MTM2013-43540-P, GVA Prometeo II/2013/013 and GVA ACOMP/2015/186 (Spain).Albanese, AA.; Bonet Solves, JA.; Ricker, WJ. (2016). Dynamics and spectrum of the Cesàro operator on C-infinity(R+). Monatshefte für Mathematik. 181:267-283. https://doi.org/10.1007/s00605-015-0863-zS267283181Albanese, 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.: Mean ergodic semigroups of operators. Rev. R. Acad. Cien. Serie A Mat. RACSAM 106, 299–319 (2012)Albanese, A.A., Bonet, J., Ricker, W.J.: Montel resolvents and uniformly mean ergodic semigroups of linear operators. Quaest. Math. 36, 253–290 (2013)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)Albanese, A.A., Bonet, J., Ricker, W.J.: Uniform mean ergodicity of $$C_0$$ C 0 -semigroups in a class of in Fréchet spaces. Funct. Approx. Comment. Math. 50, 307–349 (2014)Albanese, A.A., Bonet, J., Ricker, W.J.: On the continuous Cesàro operator in certain function spaces. Positivity 19, 659–679 (2015)Albanese, A.A., Bonet, J., Ricker, W.J.: The Cesàro operator in the Fréchet spaces $$\ell ^{p+}$$ ℓ p + and $$L^{p-}$$ L p - . Glasgow Math. J. (accepted)Arendt, W.: Gaussian estimates and interpolation of the spectrum in $$L^p$$ L p . Diff. Int. Equ. 7, 1153–1168 (1994)Bayart, F., Matheron, E.: Dynamics of linear operators. Cambridge Tracts in Mathematics, vol. 179. Cambridge University Press, Cambridge (2009)Boyd, D.W.: The spectrum of the Cesàro operator. Acta Sci. Math. (Szeged) 29, 31–34 (1968)Grosse-Erdmann, K.G., Manguillot, A.P.: Linear chaos. Universitext, Springer Verlag, London (2011)Hille, E.: Remarks on ergodic theorems. Trans. Am. Math. Soc. 57, 246–269 (1945)Jarchow, H.: Locally convex spaces. Teubner, Stuttgart (1981)Komura, T.: Semigroups of operators in locally convex spaces. J. Funct. Anal. 2, 258–296 (1968)Lin, M.: On the uniform ergodic theorem. Proc. Am. Math. Soc. 43, 337–340 (1974)Malgrange, B.: Idéaux de fonctions différentiables et division des distributions. Distributions, Editions École Polytechnique, Palaiseau, pp. 1–21 (2003)Meise, R., Vogt, D.: Introduction to functional analysis. Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press. Oxford University Press, New York (1997)Seeley, R.T.: Extension of $$C^\infty$$ C ∞ functions defined in a half space. Proc. Am. Math. Soc. 15, 625–626 (1964)Siskakis, A.G.: Composition semigroups and the Cesàro operator. J. London Math. Soc. (2) 36, 153–164 (1987)Yosida, K.: Functional analysis. Springer, New York, Berlin, Heidelberg (1980)Valdivia, M.: Topics in locally convex spaces. North-Holland Math. Stud. 67, North-Holland, Amsterdam (1982

Leading school networks, hybrid leadership in action?

A range of different constructs are used to describe and define the way that leadership operates in education settings. This range can be presented as binary categories of leadership, in which either one, or the other form of leadership is preferred, but not both. An example of this is the contrast made between solo and distributed leadership. A more sophisticated alternative has been proposed, which is to consider leadership as a hybrid activity, one which entails a range of approaches inspired by varying ideals. Taking this ‘hybrid’ notion of leadership this article explores the nature of leadership in networks of schools. Illustrated with data from three case studies of school networks this article highlights some of the issues and tensions in the enactment of the hybrid forms of leadership encountered in these networks. This article concludes with some reflections on the adoption of hybrid notions of leadership in researching and enacting educational leadership and specifically on the place of school networks in that consideration