Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces

[EN] Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.J. Bonet was partially supported by the research projects MTM2016-76647-P and GV Prometeo 2017/102 (Spain). M. Worku is supported by ISP project, Addis Ababa University, Ethiopia.Bonet Solves, JA.; Mengestie, T.; Worku, M. (2019). Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces. Results in Mathematics. 74(4):1-15. https://doi.org/10.1007/s00025-019-1123-7S115744Abanin, A.V., Tien, P.T.: Differentiation and integration operators on weighted Banach spaces of holomorphic functions. Math. Nachr. 290(8–9), 1144–1162 (2017)Atzmon, A., Brive, B.: Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, Bergman spaces and related topics in complex analysis, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, pp. 27–39 (2006)Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Math, vol. 179. Cambridge Univ. Press, Cambridge (2009)Bermúdez, T., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Austral. Math. Soc. 70, 45–54 (2004)Beltrán, M.J.: Dynamics of differentiation and integration operators on weighted space of entire functions. Studia Math. 221, 35–60 (2014)Beltrán, M.J., Bonet, J., Fernández, C.: Classical operators on weighted Banach spaces of entire functions. Proc. Am. Math. Soc. 141, 4293–4303 (2013)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonet, J.: Dynamics of the differentiation operator on weighted spaces of entire functions. Math. Z. 26, 649–657 (2009)Bonet, J.: The spectrum of Volterra operators on weighted Banach spaces of entire functions. Q. J. Math. 66, 799–807 (2015)Bonet, J., Bonilla, A.: Chaos of the differentiation operator on weighted Banach spaces of entire functions. Complex Anal. Oper. Theory 7, 33–42 (2013)Bonet, J., Taskinen, J.: A note about Volterra operators on weighted Banach spaces of entire functions. Math. Nachr. 288, 1216–1225 (2015)Constantin, O., Persson, A.-M.: The spectrum of Volterra-type integration operators on generalized Fock spaces. Bull. Lond. Math. Soc. 47, 958–963 (2015)Constantin, O., Peláez, J.-Á.: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces. J. Geom. Anal. 26, 1109–1154 (2016)De La Rosa, M., Read, C.: A hypercyclic operator whose direct sum is not hypercyclic. J. Oper. Theory 61, 369–380 (2009)Dunford, N.: Spectral theory. I. Convergence to projections. Trans. Am. Math. Soc. 54, 185–217 (1943)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear Chaos. Springer, New York (2011)Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)Lyubich, Yu.: Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition. Studia Mathematica 143(2), 153–167 (1999)Mengestie, T.: A note on the differential operator on generalized Fock spaces. J. Math. Anal. Appl. 458(2), 937–948 (2018)Mengestie, T.: Spectral properties of Volterra-type integral operators on Fock–Sobolev spaces. J. Kor. Math. Soc. 54(6), 1801–1816 (2017)Mengestie, T.: On the spectrum of volterra-type integral operators on Fock–Sobolev spaces. Complex Anal. Oper. Theory 11(6), 1451–1461 (2017)Mengestie, T., Ueki, S.: Integral, differential and multiplication operators on weighted Fock spaces. Complex Anal. Oper. Theory 13, 935–95 (2019)Mengestie, T., Worku, M.: Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces. Integr. Transf. Spec. Funct. 30, 41–54 (2019)Nagy, B., Zemanek, J.A.: A resolvent condition implying power boundedness. Studia Math. 134, 143–151 (1999)Nevanlinna, O.: Convergence of iterations for linear equations. Lecture Notes in Mathematics. ETH Zürich, Birkhäuser, Basel (1993)Ritt, R.K.: A condition that $$\lim _{n\rightarrow \infty } n^{-1}T^n =0$$. Proc. Am. Math. Soc. 4, 898–899 (1953)Ueki, S.: Characterization for Fock-type space via higher order derivatives and its application. Complex Anal. Oper. Theory 8, 1475–1486 (2014)Yosida, K.: Functional Analysis. Springer, Berlin (1978)Yosida, K., Kakutani, S.: Operator-theoretical treatment of Marko’s process and mean ergodic theorem. Ann. Math. 42(1), 188–228 (1941

Frequent hypercyclicity, chaos, and unconditional Schauder decompositions

We prove that if X is any complex separable infinite-dimensional Banach space with an unconditional Schauder decomposition, X supports an operator T which is chaotic and frequently hypercyclic. This result is extended to complex Frechet spaces with a continuous norm and an unconditional Schauder decomposition, and also to complex Frechet spaces with an unconditional basis, which gives a partial positive answer to a problem posed by Bonet. We also solve a problem of Bes and Chan in the negative by presenting hypercyclic, but non-chaotic operators on \mathbb{C}^\mathbb{N}. We extend the main result to C_0-semigroups of operators. Finally, in contrast with the complex case, we observe that there are real Banach spaces with an unconditional basis which support no chaotic operator.This work was partially supported by ANR-Projet Blanc DYNOP, by the MEC and FEDER Projects MTM2007-64222 and MTM2010-14909, and by Generalitat Valenciana Project PROMETEO/2008/101.De La Rosa Penilla, M.; Frerick, L.; Grivaux, S.; Peris Manguillot, A. (2012). Frequent hypercyclicity, chaos, and unconditional Schauder decompositions. Israel Journal of Mathematics. 190(1):389-399. https://doi.org/10.1007/s11856-011-0210-6S3893991901S. Ansari, Existence of hypercyclic operators on topological vector spaces, Journal of Functional Analysis 148 (1997), 384–390.F. Bayart and S. Grivaux, Frequently hypercyclic operators, Transactions of the American Mathematical Society 358 (2006), 5083–5117.F. Bayart and S. Grivaux, Hypercyclicity and unimodular point spectrum, Journal of Functional Analysis 226 (2005), 281–300.F. Bayart and S. Grivaux, Invariant Gaussian measures for linear operators on Banach spaces and linear dynamics, Proceedings of the London Mathematical Society 94 (2007), 181–210.F. Bayart and É. Matheron, Dynamics of Linear Operators, Cambridge University Press, Cambridge, 2009.L. Bernal-González, On hypercyclic operators on Banach spaces, Proceedings of the American Mathematical Society 127 (1999), 1003–1010.J. Bès and A. Peris, Hereditarily hypercyclic operators, Journal of Functioanl Analysis 167 (1999), 94–112.J. Bonet, F. Martínez-Giménez and A. Peris, A Banach space wich admits no chaotic operator, The Bulletin of the London Mathematical Society 33 (2001), 196–198.M. De la Rosa, L. Frerick, S. Grivaux and A. Peris, Chaos on Fréchet spaces with unconditional basis, preprint.W. T. Gowers, A solution to Banach’s hyperplane problem, The Bulletin of the London Mathematical Society 26 (1994), 523–530.W. T. Gowers and B. Maurey, Banach spaces with small spaces of operators, Mathematische Annalen 307 (1997), 543–568.W. T. Gowers and B. Maurey, The unconditional basic sequence problem, Journal of the American Mathematical Society 6 (1993), 851–874.S. Grivaux, A new class of frequently hypercyclic operators, Indiana University Mathematics Journal, to appear.K. G. Grosse-Erdmann and A. Peris, Linear Chaos, Springer-Verlag, Berlin, 2011.K. B. Laursen and M. M. Neumann, An Introduction to Local Spectral Theory, London Mathematical Society Monographs, New Series, Vol. 20, Clarendon Press, Oxford, 2000.S. Shkarin, On the spectrum of frequently hypercyclic operators, Proceedings of the American Mathematical Society 137 (2009), 123–134

Improved calibration procedures for the EM27/SUN spectrometers of the COllaborative Carbon Column Observing Network (COCCON)

In this study, an extension on the previously reported status of the COllaborative Carbon Column Observing Network\u27s (COCCON) calibration procedures incorporating refined methods is presented. COCCON is a global network of portable Bruker EM27/SUN FTIR spectrometers for deriving column-averaged atmospheric abundances of greenhouse gases. The original laboratory open-path lamp measurements for deriving the instrumental line shape (ILS) of the spectrometer from water vapour lines have been refined and extended to the secondary detector channel incorporated in the EM27/SUN spectrometer for detection of carbon monoxide (CO). The refinements encompass improved spectroscopic line lists for the relevant water lines and a revision of the laboratory pressure measurements used for the analysis of the spectra. The new results are found to be in good agreement with those reported by Frey et al. (2019) and discussed in detail. In addition, a new calibration cell for ILS measurements was designed, constructed and put into service. Spectrometers calibrated since January 2020 were tested using both methods for ILS characterization, open-path (OP) and cell measurements. We demonstrate that both methods can detect the small variations in ILS characteristics between different spectrometers, but the results of the cell method indicate a systematic bias of the OP method. Finally, a revision and extension of the COCCON network instrument-to-instrument calibration factors for XCO2, XCO and XCH4 is presented, incorporating 47 new spectrometers (of 83 in total by now). This calibration is based on the reference EM27/SUN spectrometer operated by the Karlsruhe Institute of Technology (KIT) and spectra collected by the collocated TCCON station Karlsruhe. Variations in the instrumental characteristics of the reference EM27/SUN from 2014 to 2017 were detected, probably arising from realignment and the dual-channel upgrade performed in early 2018. These variations are considered in the evaluation of the instrument-specific calibration factors in order to keep all tabulated calibration results consistent

Plasma enhanced oxidation of nickel silicides at low temperature

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Transplantation of human chromaffin cells for control of intractable cancer pain

Adrenal medullary chromaffin cells produce high levels of endogenous opioid peptides. Recent data suggest that transplantation injected locally into the spinal subarachnoid space reduced intractable malignant pain. In order to determine the feasibility, the efficacy and the risks of using adrenal medullary tissue for control of irreducible pain, we have developed a transplantation protocol on cancer pain patients selected when they required chronic intrathecal injection of morphine and progressively increasing doses to maintain the level of analgesic effects. At the present time, our clinical trial involves 8 patients. We report here our initial results (mean follow-up: 5 months). The various data collected before and after the intrathecal administration of chromaffin cells included: 1) Pain evaluation over time, with concomitant narcotic intake, 2) CSF sampling through an implanted access port to determine the following biological parameters: biochemical assay for opioid peptides, cell count and phenotyping of lymphocytes, 3) peripheral blood samples for lymphocyte typing. The results confirm the efficacy of adrenal medullary transplantation into spinal CSF for controlling irreducible cancer pain. Complementary intrathecal and oral morphine were totally stopped in 2 cases and stabilized in 5 others. It seems essential to have an important volume of grafted tissue to achieve analgesia with high levels of metenkephalin in CSF. A progressive decrease in metenkephalin release was observed from 2 to 4 months after the transplantation. Two patients with a long-term follow-up (8 and 12 months) needed another intrathecal chromaffin cell graft

Invariants of Automatic Presentations and Semi−Synchronous Transductions

Automatic structures are countable structures finitely presentable by a collection of automata. We study questions related to properties invariant with respect to the choice of an automatic presentation. We give a negative answer to a question of Rubin concerning definability of intrinsically regular relations by showing that order-invariant first-order logic can be stronger than first-order logic with counting on automatic structures. We introduce a notion of equivalence of automatic presentations, define semi-synchronous transductions, and show how these concepts correspond. Our main result is that a one-to-one function on words preserves regularity as well as non-regularity of all relations iff it is a semi-synchronous transduction. We also characterize automatic presentations of the complete structures of Blumensath and Gr�del