Inner Derivations and Weak-2-Local Derivations on the C*-Algebra C-0(L, A)

[EN] Let L be a locally compact Hausdorff space. Suppose A is a -algebra with the property that every weak-2-local derivation on A is a (linear) derivation. We prove that every weak-2-local derivation on is a (linear) derivation. Among the consequences we establish that if B is an atomic von Neumann algebra or a compact -algebra, then every weak-2-local derivation on is a linear derivation. We further show that, for a general von Neumann algebra M, every 2-local derivation on is a linear derivation. We also prove several results representing derivations on and on as inner derivations determined by multipliers.Jorda Mora, E.; Peralta, A. (2017). Inner Derivations and Weak-2-Local Derivations on the C*-Algebra C-0(L, A). Integral Equations and Operator Theory. 89(1):89-110. https://doi.org/10.1007/s00020-017-2390-xS89110891Akemann, C.A., Elliott, G.E., Pedersen, G.K., Tomiyama, J.: Derivations and multipliers of $$\text{ C }^*$$ C ∗ -algebras. Am. J. Math. 98(3), 679–708 (1976)Akemann, C.A., Johnson, B.E.: Derivations of non-separable $$\text{ C }^*$$ C ∗ -algebras. J. Funct. Anal. 33, 311–331 (1979)Akemann, C.A., Pedersen, G.K., Tomiyama, J.: Multipliers of $$\text{ C }^*$$ C ∗ -algebras. J. Funct. Anal. 13, 277–301 (1973)Archbold, R.J.: On the norm of an inner derivation of a $$\text{ C }^*$$ C ∗ -algebra. Math. Proc. Camb. Philos. Soc. 84, 273–291 (1978)Archbold, R.J., Somerset, D.W.B.: Inner derivations and primal ideals of $$\text{ C }^*$$ C ∗ -algebras. II. Proc. Lond. Math. Soc. (3) 88(1), 225–250 (2004)Ayupov, S., Arzikulov, F.N.: 2-Local derivations on algebras of matrix-valued functions on a compact, preprint (2015). arXiv:1509.05701v1Ayupov, Sh, Kudaybergenov, K.K.: $$2$$ 2 -local derivations on von Neumann algebras. Positivity 19(3), 445–455 (2015). doi: 10.1007/s11117-014-0307-3Burgos, M., Cabello, J.C., Peralta, A.M.: Weak-local triple derivations on $$\text{ C }^*$$ C ∗ -algebras and JB $$^*$$ ∗ -triples. Linear Algebra Appl. 506, 614–627 (2016). doi: 10.1016/j.laa.2016.06.042Cabello, J.C., Peralta, A.M.: Weak-2-local symmetric maps on $$\text{ C }^*$$ C ∗ -algebras. Linear Algebra Appl. 494, 32–43 (2016). doi: 10.1016/j.laa.2015.12.024Cabello, J.C., Peralta, A.M.: On a generalized Šemrl’s theorem for weak-2-local derivations on $$B(H)$$ B ( H ) . Banach J. Math. Anal. 11(2), 382–397 (2017)Elliott, G.A.: Some $$\text{ C }^*$$ C ∗ -algebras with outer derivations. III. Ann. Math. (2) 106(1), 121–143 (1977)Elliott, G.A.: On derivations of AW $$^*$$ ∗ -algebras. Tohoku Math. J. 30, 263–276 (1978)Essaleh, A.B.A., Peralta, A.M., Ramírez, M.I.: Weak-local derivations and homomorphisms on $$\text{ C }^*$$ C ∗ -algebras. Linear Multilinear Algebra 64(2), 169–186 (2016)Essaleh, A.B.A., Peralta, A.M., Ramírez, M.I.: CORRIGENDUM: Weak-local derivations and homomorphisms on $$\text{ C }^*$$ C ∗ -algebras. Linear Multilinear Algebra 64(5), 1009–1010 (2016)Gajendragadkar, P.: Norm of a derivation of a von Neumann algebra. Trans. Am. Math. Soc. 170, 165–170 (1972)Gogić, I.: Derivations of subhomogeneous $$\text{ C }^*$$ C ∗ -algebras are implemented by local multipliers. Proc. Am. Math. Soc. 141(11), 3925–3928 (2013)Gogić, I.: The local multiplier algebra of a $$\text{ C }^*$$ C ∗ -algebra with finite dimensional irreducible representations. J. Math. Anal. Appl. 408(2), 789–794 (2013)Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis. Vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Springer, New York, Berlin (1970)Jordá, E., Peralta, A.M.: Stability of derivations under weak-2-local continuous perturbations. Aequationes Math. 91, 99–114 (2017)Kadison, R.V., Lance, E.C., Ringrose, J.R.: Derivations and automorphisms of operator algebras II. J. Funct. Anal. 1, 204–221 (1967)Kyle, J.: Norms of derivations. J. Lond. Math. Soc. II Ser. 16, 297–312 (1977)Kowalski, S., Słodkowski, Z.: A characterization of multiplicative linear functionals in Banach algebras. Stud. Math 67, 215–223 (1980)Lance, E.C.: Automorphisms of certain operator algebras. Am. J. Math. 91, 160–174 (1969)Michael, E.: Continuous selections I. Ann. Math. 63, 361–382 (1956)Niazi, M., Peralta, A.M.: Weak-2-local derivations on $$\mathbb{M}_n$$ M n . FILOMAT 31(6), 1687–1708 (2017)Niazi, M., Peralta, A.M.: Weak-2-local $$^*$$ ∗ -derivations on $$B(H)$$ B ( H ) are linear $$^*$$ ∗ -derivations. Linear Algebra Appl. 487, 276–300 (2015)Pedersen, G.K.: Approximating derivations on ideals of $$\text{ C }^*$$ C ∗ -algebras. Invent. Math. 45, 299–305 (1978)Ringrose, J.R.: Automatic continuity of derivations of operator algebras. J. Lond. Math. Soc. 2(5), 432–438 (1972)Sakai, S.: On a conjecture of Kaplansky. Tohoku Math. J. 12, 31–33 (1960)Sakai, S.: $$\text{ C }^*$$ C ∗ -Algebras and $$W^*$$ W ∗ -Algebras. Springer, Berlin (1971)Sakai, S.: Derivations of simple $$\text{ C }^*$$ C ∗ -algebras. II. Bull. Soc. Math. Fr. 99, 259–263 (1971)Šemrl, P.: Local automorphisms and derivations on $$B(H)$$ B ( H ) . Proc. Am. Math. Soc. 125, 2677–2680 (1997)Somerset, D.W.B.: The inner derivations and the primitive ideal space of a $$\text{ C }^*$$ C ∗ -algebra. J. Oper. Theory 29, 307–321 (1993)Somerset, D.W.B.: Inner derivations and primal ideals of $$\text{ C }^*$$ C ∗ -algebras. J. Lond. Math. Soc. 2(50), 568–580 (1994)Somerset, D.W.B.: The local multiplier algebra of a $$\text{ C }^*$$ C ∗ -algebra. II. J. Funct. Anal. 171(2), 308–330 (2000)Stampfli, J.G.: The norm of a derivation. Pac. J. Math. 33(3), 737–747 (1970)Takesaki, M.: Theory of Operator Algebras I. Springer, Berlin (1979)Zsido, L.: The norm of a derivation in a W $$^*$$ ∗ -algebra. Proc. Am. Math. Soc. 38, 147–150 (1973

Stability of derivations under weak-2-local continuous perturbations

[EN] Let ¿ be a compact Hausdorff space and let A be a C¿ -algebra. We prove that if every weak-2-local derivation on A is a linear derivation and every derivation on C(¿, A) is inner, then every weak-2-local derivation ¿ : C(¿, A) ¿ C(¿, A) is a (linear) derivation. As a consequence we derive that, for every complex Hilbert space H, every weak-2-local derivation ¿ : C(¿, B(H)) ¿ C(¿, B(H)) is a (linear) derivation. We actually show that the same conclusion remains true when B(H) is replaced with an atomic von Neumann algebra. With a modified technique we prove that, if B denotes a compact C¿ -algebra (in particular, when B = K(H)), then every weak-2-local derivation on C(¿, B) is a (linear) derivation. Among the consequences, we show that for each von Neumann algebra M and every compact Hausdorff space ¿, every 2-local derivation on C(¿, M) is a (linear) derivation.E. Jorda is partially supported by the Spanish Ministry of Economy and Competitiveness Project MTM2013-43540-P and Generalitat Valenciana Grant AICO/2016/054. A. M. Peralta is partially supported by the Spanish Ministry of Economy and Competitiveness and European Regional Development Fund Project No. MTM2014-58984-P and Junta de Andalucia Grant FQM375.Jorda Mora, E.; Peralta, AM. (2017). Stability of derivations under weak-2-local continuous perturbations. Aequationes Mathematicae. 91(1):99-114. https://doi.org/10.1007/s00010-016-0438-7S99114911Akemann C.A., Johnson B.E.: Derivations of non-separable C*-algebras. J. Funct. Anal. 33, 311–331 (1979)Alexander J.: Compact Banach algebras. Proc. London Math. Soc. 18, 1–18 (1968)Aupetit B.: A Primer on Spectral Theory (Universitext). Springer, New York (1991)Ayupov, Sh., Arzikulov, F.N.: 2-Local derivations on algebras of matrix-valued functions on a compact. (2015) (preprint) arXiv:1509.05701v1Ayupov Sh., Kudaybergenov K.K.: 2-local derivations on von Neumann algebras. Positivity 19(3), 445–455 (2015) doi: 10.1007/s11117-014-0307-3Cabello J.C., Peralta A.M.: Weak-2-local symmetric maps on C*-algebras. Linear Algebra Appl. 494, 32–43 (2016) doi: 10.1016/j.laa.2015.12.024Cabello, J.C., Peralta, A.M.: On a generalized Šemrl’s theorem for weak-2-local derivations on B(H). Banach J. Math. Anal. (to appear) arXiv:1511.07987v2Essaleh A.B.A., Peralta A.M., Ramírez M.I.: Weak-local derivations and homomorphisms on C*-algebras. Linear Multilinear Algebra 64(2), 169–186 (2016). doi: 10.1080/03081087.2015.1028320Johnson, B.E.: Cohomology in Banach algebras, vol. 127. Memoirs of the American Mathematical Society, Providence (1972)Johnson B.E.: Local derivations on C*-algebras are derivations. Trans. Amer. Math. Soc. 353, 313–325 (2001)Kadison R.V.: Derivations of operator algebras. Ann. Math. 83(2), 280–293 (1966)Kadison R.V.: Local derivations. J. Algebra 130, 494–509 (1990)Kadison R.V., Lance E.C., Ringrose J.R.: Derivations and automorphisms of operator algebras II. J. Funct. Anal. 1, 204–221 (1947)Niazi M., and Peralta, A.M.: Weak-2-local derivations on $${\mathbb{M}_n}$$ M n . FILOMAT (to appear)Niazi M., Peralta A.M.: Weak-2-local *-derivations on B(H) are linear *-derivations. Linear Algebra Appl. 487, 276–300 (2015)Ringrose J.R.: Automatic continuity of derivations of operator algebras. J. London Math. Soc. (2) 5, 432–438 (1972)Runde, V.: Lectures on Amenability. Lecture Notes in Mathematics, vol. 1774. Springer, Berlin (2002)Sakai S.: On a conjecture of Kaplansky. Tohoku Math. J. 12, 31–33 (1960)Sakai S.: C*-algebras and W*-algebras. Springer, Berlin (1971)Šemrl P.: Local automorphisms and derivations on B(H). Proc. Amer. Math. Soc. 125, 2677–2680 (1997)Stampfli J.G.: The norm of a derivation. Pac. J. Math. 33(3), 737–747 (1970)Takesaki M.: Theory of operator algebras I. Springer, Berlin (1979

Weighted vector-valued holomorphic functions on Banach spaces

We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a function f defined in a subset A of an open and connected subset U of a Banach space X, with values in another Banach space X, and admitting certain weak extensions in a Banach space of holomorphic functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.The author wants to thank J. Bonet for several references, discussions, and ideas provided, which were very helpful and in particular allowed him to prove Theorem 7, Proposition 8, and Examples 15 and 16. Remark 4 is due to him. The participation of M. J. Beltran in a lot of discussions during all the work has also been very important. Her ideas are also reflected in the paper. The author is also indebted to L. Frerick and J. Wengenroth for communicating to him Lemma 2. The remarks and corrections of the referee have been also really helpful to the final version. The author thanks him/her for that. This research was partially supported by MEC and FEDER Project MTM2010-15200, GV Project ACOMP/2012/090, and Programa de Apoyo a la Investigacin y Desarrollo de la UPV PAID-06-12.Jorda Mora, E. (2013). Weighted vector-valued holomorphic functions on Banach spaces. Abstract and Applied Analysis. 2013:1-9. https://doi.org/10.1155/2013/501592192013Dunford, N. (1938). Uniformity in Linear Spaces. Transactions of the American Mathematical Society, 44(2), 305. doi:10.2307/1989974Bogdanowicz, W. M. (1969). Analytic continuation of holomorphic functions with values in a locally convex space. Proceedings of the American Mathematical Society, 22(3), 660-660. doi:10.1090/s0002-9939-1969-0250067-1Arendt, W., & Nikolski, N. (2000). Vector-valued holomorphic functions revisited. Mathematische Zeitschrift, 234(4), 777-805. doi:10.1007/s002090050008Bonet, J., Frerick, L., & Jordá, E. (2007). Extension of vector-valued holomorphic and harmonic functions. Studia Mathematica, 183(3), 225-248. doi:10.4064/sm183-3-2Frerick, L., Jordá, E., & Wengenroth, J. (2009). Extension of bounded vector-valued functions. Mathematische Nachrichten, 282(5), 690-696. doi:10.1002/mana.200610764GROSSE-ERDMANN, K.-G. (2004). A weak criterion for vector-valued holomorphy. Mathematical Proceedings of the Cambridge Philosophical Society, 136(2), 399-411. doi:10.1017/s0305004103007254Laitila, J., & Tylli, H.-O. (2006). Composition operators on vector-valued harmonic functions and Cauchy transforms. Indiana University Mathematics Journal, 55(2), 719-746. doi:10.1512/iumj.2006.55.2785Beltrán, M. J. (2011). Linearization of weighted (LB)-spaces of entire functions on Banach spaces. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 106(2), 275-286. doi:10.1007/s13398-011-0049-zCarando, D., & Zalduendo, I. (2004). Linearization of functions. Mathematische Annalen, 328(4), 683-700. doi:10.1007/s00208-003-0502-1Mujica, J. (1991). Linearization of Bounded Holomorphic Mappings on Banach Spaces. Transactions of the American Mathematical Society, 324(2), 867. doi:10.2307/2001745Fabian, M., Habala, P., Hájek, P., Montesinos, V., & Zizler, V. (2011). Banach Space Theory. CMS Books in Mathematics. doi:10.1007/978-1-4419-7515-7Dineen, S. (1999). Complex Analysis on Infinite Dimensional Spaces. Springer Monographs in Mathematics. doi:10.1007/978-1-4471-0869-6Boyd, C., & Lassalle, S. (2009). GEOMETRY AND ANALYTIC BOUNDARIES OF MARCINKIEWICZ SEQUENCE SPACES. The Quarterly Journal of Mathematics, 61(2), 183-197. doi:10.1093/qmath/han037Globevnik, J. (1978). On interpolation by analytic maps in infinite dimensions. Mathematical Proceedings of the Cambridge Philosophical Society, 83(2), 243-252. doi:10.1017/s0305004100054505Globevnik, J. (1979). Boundaries for polydisc algebras in infinite dimensions. Mathematical Proceedings of the Cambridge Philosophical Society, 85(2), 291-303. doi:10.1017/s0305004100055705Seip, K. (1993). Beurling type density theorems in the unit disk. Inventiones Mathematicae, 113(1), 21-39. doi:10.1007/bf01244300Ng, K. (1971). On a Theorem of Dixmier. MATHEMATICA SCANDINAVICA, 29, 279. doi:10.7146/math.scand.a-11054Bochnak, J., & Siciak, J. (1971). Polynomials and multilinear mappings in topological vector-spaces. Studia Mathematica, 39(1), 59-76. doi:10.4064/sm-39-1-59-76Gramsch, B. (1977). Ein Schwach-Stark-Prinzip der Dualit�tstheorie lokalkonvexer R�ume als Fortsetzungsmethode. Mathematische Zeitschrift, 156(3), 217-230. doi:10.1007/bf01214410Bonet, J., Gómez-Collado, M. C., Jornet, D., & Wolf, E. (2012). Operator-weighted composition operators between weighted spaces of vector-valued analytic functions. Annales Academiae Scientiarum Fennicae Mathematica, 37, 319-338. doi:10.5186/aasfm.2012.3723Bierstedt, K. D., Bonet, J., & Galbis, A. (1993). Weighted spaces of holomorphic functions on balanced domains. The Michigan Mathematical Journal, 40(2), 271-297. doi:10.1307/mmj/1029004753Bierstedt, K. D., & Summers, W. H. (1993). Biduals of weighted banach spaces of analytic functions. Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics, 54(1), 70-79. doi:10.1017/s1446788700036983Bonet, J., & Wolf, E. (2003). A note on weighted Banach spaces of holomorphic functions. Archiv der Mathematik, 81(6), 650-654. doi:10.1007/s00013-003-0568-8Aron, R. M., & Schottenloher, M. (1974). Compact holomorphic mappings on Banach spaces and the approximation property. Bulletin of the American Mathematical Society, 80(6), 1245-1250. doi:10.1090/s0002-9904-1974-13701-

Extension operators for smooth functions on compact subsets of the reals

[EN] We introduce sufficient as well as necessary conditions for a compact set K such that there is a continuous linear extension operator from the space of restrictions C-infinity (K) = {F vertical bar(K) : F is an element of C-infinity (R)} to C-infinity (R). This allows us to deal with examples of the form K = {a(n) : n is an element of N} boolean OR{0} for a(n) -> 0 previously considered by Fefferman and Ricci as well as Vogt.The research of all authors was partially supported by GVA AICO/2016/054 . The research of the second author was partially supported by the project MTM2016-76647-P.Frerick, L.; Jorda Mora, E.; Wengenroth, J. (2020). Extension operators for smooth functions on compact subsets of the reals. Mathematische Zeitschrift. 295(3-4):1537-1552. https://doi.org/10.1007/s00209-019-02388-5S153715522953-4Bos, Len P., Milman, Pierre D.: Sobolev–Gagliardo–Nirenberg and Markov type inequalities on subanalytic domains. Geom. Funct. Anal. 5(6), 853–923 (1995)Bierstone, Edward, Milman, Pierre D.: Geometric and differential properties of subanalytic sets. Ann. Math. (2) 147(3), 731–785 (1998)Bierstone, Edward, Milman, Pierre D., Pawłucki, Wiesław: Composite differentiable functions. Duke Math. J. 83(3), 607–620 (1996)DeVore, Ronald A., Lorentz, George G.: Constructive approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 303. Springer, Berlin (1993)Fefferman, Charles: $$C^m$$ extension by linear operators. Ann. Math. (2) 166(3), 779–835 (2007)Frerick, Leonhard, Jordá, Enrique, Wengenroth, Jochen: Tame linear extension operators for smooth Whitney functions. J. Funct. Anal. 261(3), 591–603 (2011)Frerick, Leonhard, Jordá, Enrique, Wengenroth, Jochen: Whitney extension operators without loss of derivatives. Rev. Mat. Iberoam. 32(2), 377–390 (2016)Fefferman, Charles, Ricci, Fulvio: Some examples of $$C^\infty$$ extension by linear operators. Rev. Mat. Iberoam. 28(1), 297–304 (2012)Frerick, Leonhard: Extension operators for spaces of infinite differentiable Whitney jets. J. Reine Angew. Math. 602, 123–154 (2007)Goncharov, Alexander: A compact set without Markov’s property but with an extension operator for $$C^\infty$$-functions. Studia Math. 119(1), 27–35 (1996)Hörmander, Lars: The analysis of linear partial differential operators. I, second ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer, Berlin, Distribution theory and Fourier analysis (1990)Malgrange, Bernard: Ideals of differentiable functions, Tata Institute of Fundamental Research Studies in Mathematics, No. 3, Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, (1967)Merrien, Jean: Prolongateurs de fonctions différentiables d’une variable réelle. J. Math. Pures Appl. (9) 45, 291–309 (1966)Mitjagin, B.S.: Approximate dimension and bases in nuclear spaces. Uspehi Mat. Nauk 16(4 (100)), 63–132 (1961)Meise, Reinhold, Vogt, Dietmar: Introduction to functional analysis, Oxford Graduate Texts in Mathematics, vol. 2. The Clarendon Press, Oxford University Press, New York (1997). Translated from the German by M. S. Ramanujan and revised by the authorsPawłucki, Wiesław: On the algebra of functions $$\mathscr {C}^k$$-extendable for each $$k$$ finite. Proc. Am. Math. Soc. 133(2), 481–484 (2005). (Electronic)Pawłucki, Wiesław, Pleśniak, Wiesław: Extension of $$C^\infty$$ functions from sets with polynomial cusps. Studia Math. 88(3), 279–287 (1988)Stein, Elias M.: Singular integrals and differentiability properties of functions, Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)Tidten, Michael: Fortsetzungen von $$C^{\infty }$$-Funktionen, welche auf einer abgeschlossenen Menge in $${ R}^{n}$$ definiert sind. Manuscripta Math. 27(3), 291–312 (1979)Vogt, Dietmar: Restriction spaces of $$A^\infty$$. Rev. Mat. Iberoam. 30(1), 65–78 (2014)Vogt, Dietmar, Wagner, Max Josef: Charakterisierung der Quotientenräume von $$s$$ und eine Vermutung von Martineau. Studia Math. 67(3), 225–240 (1980)Wengenroth, Jochen: Derived functors in functional analysis. Lecture Notes in Mathematics, vol. 1810. Springer, Berlin (2003)Whitney, Hassler: Analytic extensions of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36(1), 63–89 (1934)Whitney, Hassler: On ideals of differentiable functions. Am. J. Math. 70, 635–658 (1948

Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type

[EN] Given an affine symbol phi and a multiplier w, we focus on the weighted composition operator C-w,C-phi acting on the spaces Exp and Exp(0) of entire functions of exponential and of infraexponential type, respectively. We characterize the continuity of the operator and, for w the product of a polynomial by an exponential function, we completely characterize power boundedness and (uniform) mean ergodicity. In the case of multiples of composition operators, we also obtain the spectrum and characterize hypercyclicity.The research of the first author was supported by the research project MTM2016-76647-P. The research of the second author was partially supported by MTM2016-76647-P and PID2020-119457GB-I00Beltrán Meneu, MJ.; Jorda Mora, E. (2021). Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type. Mediterranean Journal of Mathematics. 18(5):1-18. https://doi.org/10.1007/s00009-021-01850-1S11818

A characterization of the Schur property through the disk algebra

[EN] In this paper we give a new characterization of when a Banach space E has the Schur property in terms of the disk algebra. We prove that E has the Schur property if and only if A(D, E) = A(D,E-w). (C) 2016 Elsevier Inc. All rights reserved.The first and third authors were supported by MINECO MTM2014-57838-C2-2-P and Prometeo II/2013/013. The second author was supported by MINECO MTM2013-43540-P.García, D.; Jorda Mora, E.; Maestre, M. (2017). A characterization of the Schur property through the disk algebra. Journal of Mathematical Analysis and Applications. 445(2):1310-1320. https://doi.org/10.1016/j.jmaa.2016.01.028S13101320445

Strongly continuous semigroups on some Fréchet spaces

We prove that for a strongly continuous semigroup T on the Frechet space omega of all scalar sequences, its generator is a continuous linear operator A is an element of L(omega) and that, for all x is an element of omega and t >= 0, the series exp(tA)(x) = Sigma(infinity)(k=0) t(k)/k! A(k)(x) converges to T-t(x). This solves a problem posed by Conejero. Moreover, we improve recent results of Albanese, Bonet, and Ricker about semigroups on strict projective limits of Banach spaces. (C) 2013 Elsevier Inc. All rights reserved.The research was partially done during a stay of the fourth named author at EPSA-UPV. This visit was supported by Proyecto Prometeo 11/2013/013. The research of the first and second named authors was supported by MICINN and FEDER, Project MTM2010-15200. The research of the second named author was partially supported by Programa de Apoyo a la Investigacion y Desarrollo de la UPV PAID-06-12.Frerick, L.; Jorda Mora, E.; Kalmes, T.; Wengenroth, J. (2014). Strongly continuous semigroups on some Fréchet spaces. Journal of Mathematical Analysis and Applications. 412(1):121-124. https://doi.org/10.1016/j.jmaa.2013.10.053S121124412

The division problem for tempered distributions of one variable

We give a characterization, in one variable case, of those C-infinity multipliers F such that the division problem is solvable in S'(R). For these functions F is an element of O-M (R) we even prove that the multiplication operator M-F(G) = FG has a continuous linear right inverse on S'(R), in contrast to what happens in the several variables case, as was shown by Langenbruch. (C) 2011 Elsevier Inc. All rights reserved.This research was partially supported by MEC and FEDER Project MTM2010-15200 and by GV Project Prometeo/2008/101.Bonet Solves, JA.; Frerick, L.; Jorda Mora, E. (2012). The division problem for tempered distributions of one variable. Journal of Functional Analysis. 262(5):2349-2358. https://doi.org/10.1016/j.jfa.2011.12.00623492358262

Dynamics and spectra of composition operators on the Schwartz space

[EN] In this paper we study the dynamics of the composition operators defined in the Schwartz space of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol ¿ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when ¿ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.The present research was partially supported by the projects MTM2016-76647-P, ACOMP/2015/186, Prometeo/2017/102 of the Generalitat Valenciana (Spain). The third author was partially supported by GVA, Project AICO/2016/054.Fernández Rosell, C.; Galbis Verdu, A.; Jorda Mora, E. (2018). Dynamics and spectra of composition operators on the Schwartz space. Journal of Functional Analysis. 274(12):3503-3530. https://doi.org/10.1016/j.jfa.2017.11.005S350335302741

Tingley's problem for p-Schatten von Neumann classes

[EN] Let H and H' be the complex Hilbert spaces. For p is an element of] 1, infinity[\{2} we consider the Banach space C-p(H) of all p-Schatten von Neumann operators, whose unit sphere is denoted by S(C-p(H)). In this paper we prove that every surjective isometry Delta: S(C-p(H)) -> S(C-p(H')) can be extended to a complex linear or to a conjugate linear surjective isometry T: C-p(H) -> C-p(H').The first and third authors were partially supported by the Spanish Ministry of Science, Innovation and Universities (MICINN) and European RegionalDevelopment Fund project no. PGC2018-093332-B-I00, Programa Operativo FEDER 2014-2020 and Consejeria de Economia y Conocimiento de la Junta de Andalucia grant number A-FQM-242-UGR18, and Junta de Andalucia grant FQM375. The second author was partially supported by the project MTM2016-76647-P.Fernández-Polo, FJ.; Jorda Mora, E.; Peralta, AM. (2020). Tingley's problem for p-Schatten von Neumann classes. Journal of Spectral Theory. 10(3):809-841. https://doi.org/10.4171/JST/313S80984110