# zbMATH — the first resource for mathematics

Schauder decompositions and the Grothendieck and Dunford–Pettis properties in Köthe echelon spaces of infinite order. (English) Zbl 1131.46005
A Fréchet space $$X$$ is called a Grothendieck space if every sequence in the dual space $$X'$$ which is convergent for the weak* topology $$\sigma(X',X)$$ is also convergent for the weak topology $$\sigma(X',X'')$$. A Fréchet space $$X$$ is said to have the Dunford–Pettis property if every element of $$L(X,Y)$$, for $$Y$$ an arbitrary quasicomplete locally convex Hausdorff space ($$L(X,Y)$$ denotes the space of all linear continuous maps from $$X$$ into $$Y$$), which transforms bounded subsets of $$X$$ into relatively weakly compact subsets of $$Y$$, also transforms weakly compact subsets of $$X$$ into compact subsets of $$Y$$. The class of Fréchet spaces which are Grothendieck spaces with the Dunford–Pettis property are denoted by GDP-spaces. Well-known examples of GDP-spaces include $$L^\infty$$, $$H^\infty ( D)$$, injective Banach spaces, and certain $$C(K)$$ spaces.
In the setting of non-normable Fréchet spaces, the only known examples of GDP-spaces were given by Fréchet–Montel spaces. Other than Montel spaces, the authors show that every Köthe echelon space $$\lambda_\infty(A)$$, with $$A$$ an arbitrary Köthe matrix, is a GDP-space. In the case when $$\lambda_\infty(A)\not =\lambda_0(A)$$ (note that $$\lambda_\infty(A) =\lambda_0(A)$$ if and only if $$\lambda_\infty(A)$$ is Montel), the authors show that $$\lambda_\infty(A)$$ still admits unconditional Schauder decompositions, provided that it satisfies the density condition. This contrasts the situation for GDP-Banach spaces which do not have any Schauder decomposition by a result of D. Dean [Proc. Am. Math. Soc. 18, 619–623 (1967; Zbl 0158.13503)]. Consequences for spectral measures are also given.

##### MSC:
 46A45 Sequence spaces (including Köthe sequence spaces) 46A04 Locally convex Fréchet spaces and (DF)-spaces 46G10 Vector-valued measures and integration 47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) 46A35 Summability and bases in topological vector spaces
Full Text:
##### References:
 [1] A.A. Albanese, Montel subspaces of Fréchet spaces of Moscatelli type. Glasgow Math. J. 39 (1997), 345–350. · Zbl 0923.46011 [2] C.D. Aliprantis, O. Burkinshaw, Locally Solid Riesz Spaces. Academic Press, New York–San Francisco, 1978. · Zbl 0402.46005 [3] F. Bastin, Dinstinguishedness of weighted Fréchet spaces of continuous functions. Proc. Edinburgh Math. Soc. (2) 35 (1992), 271–283. · Zbl 0738.46006 [4] C. Bessaga, A.A. Pelczyński, S. Rolewicz, On diametrical approximative dimension and linear homogeneity of F-spaces. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 9 (1961), 677–683. · Zbl 0109.33502 [5] K.D. Bierstedt, J. Bonet, Stefan Heinrich’s density condition for Fréchet spaces and the characterization of distinguished Köthe echelon spaces. Math. Nachr. 135 (1988), 149–180. · Zbl 0688.46001 [6] K.D. Bierstedt, J. Bonet, Some aspects of the modern theory of Fréchet spaces. Rev. R. Acad. Cien. Serie A Mat. RACSAM 97 (2003), 159–188. · Zbl 1085.46001 [7] K.D. Bierstedt, R.G. Meise, W.H. Summers, Köthe sets and Köthe sequence spaces, In: Functional Analysis. Holomorphy and Approximation Theory (Rio de Janeiro, 1980), North Holland Math. Stud. 71 (1982), 27–91. [8] J. Bonet, M. Lindström, Convergent sequences in duals of Fréchet spaces. pp. 391–404 in “Functional Analysis”, Proc. of the Essen Conference, Marcel Dekker, New York, 1993. · Zbl 0804.46003 [9] J. Bonet, W.J. Ricker, The canonical spectral measure in Köthe echelon spaces. Integral Equations Operator Theory, 53 (2005), 477–496. · Zbl 1109.46047 [10] D.W. Dean, Schauder decompositions in (m). Proc. Amer. Math. Soc. 18 (1967), 619–623. · Zbl 0158.13503 [11] J.C. Díaz, C. Fernández, On quotients of sequence spaces of infinite order. Archiv Math. (Basel), 66 (1996), 207–213. [12] J.C. Díaz, G. Metafune, The problem of topologies of Grothendieck for quojections. Result. Math. 21 (1992), 299–312. · Zbl 0787.46002 [13] J.C. Díaz, M.A. Minarno, On total bounded sets in Köthe echelon spaces. Bull. Soc. Roy. Sci. Liege 59 (1990), 483–492. · Zbl 0759.46004 [14] J. Diestel, A survey of results related to the Dunford-Pettis property. Contemp. Math. 2 (Amer. Math. Soc., 1980), pp.15–60. [15] J. Diestel, J.J.Jr. Uhl, Vector Measures. Math. Surveys No. 15, Amer. Math. Soc., Providence, 1977. [16] R.E. Edwards, Functional Analysis. Reinhart and Winston, New York, 1965. · Zbl 0182.16101 [17] A. Fernández, F. Naranjo, Nuclear Fréchet lattices. J. Austral. Math. Soc. 72 (2002), 409–417. · Zbl 1031.46007 [18] N.J. Kalton, Schauder decompositions in locally convex spaces. Math Proc. Cambridge Phil. Soc. 68 (1970), 377–392. · Zbl 0196.13505 [19] G. Köthe, Topological Vector Spaces I (2nd Edition: revised). Springer Verlag, Berlin-Heidelberg-New York, 1983. [20] H.P. Lotz, Tauberian theorems for operators on Land similar spaces. pp. 117–133 in “Functional Analysis Surveys and Recent Results”, North Holland, Amsterdam, 1984. [21] H.P. Lotz, Uniform convergence of operators on Land similar spaces. Math. Z. 190 (1985), 207–220. · Zbl 0623.47033 [22] J.T. Marti, Introduction to the Theory of Bases. Springer Verlag, Berlin, 1969. · Zbl 0191.41301 [23] R.G. Meise, D. Vogt, Introduction to Functional Analysis. Clarendon Press, Oxford, 1997. · Zbl 0924.46002 [24] S. Okada, Spectrum of scalar-type spectral operators and Schauder decompositions. Math. Nachr. 139 (1988), 167–174. · Zbl 0682.47018 [25] W.J. Ricker, Spectral operators of scalar-type in Grothendieck spaces with the Dunford-Pettis property. Bull. London Math. Soc, 17 (1985), 268–270. · Zbl 0584.47033 [26] W.J. Ricker, Countable additivity of multiplicative, operator-valued set functions. Acta Math. Hung. 47 (1986), 121–126. · Zbl 0615.47028 [27] W.J. Ricker, Spectral measures, boundedly {$$\sigma$$}-complete Boolean algebras and applications to operator theory. Trans. Amer. Math. Soc. 304 (1987), 819–838. · Zbl 0642.47029 [28] W.J. Ricker, Operator algebras generated by Boolean algebras of projections in Montel spaces. Integral Equations Operator Theory, 12 (1989), 143–145. · Zbl 0681.47021 [29] W.J. Ricker, Resolutions of the identity in Fréchet spaces. Integral Equations Operator Theory, 41 (2001), 63–73. · Zbl 0995.46027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.