Fixed point theory in Fréchet spaces for Volterra type operators

New fixed point theorems for maps (single and multivalued) between Fréchet spaces are presented. The proof relies on fixed point theory in Banach spaces and viewing a Fréchet space as the projective limit of a sequence of Banach spaces.     

Double convergence and products of Fréchet spaces

The paper is devoted to convergence of double sequences and its application to products. In a convergence space we recognize three types of double convergences and points, respectively. We give examples and describe their structure and properties. We investigate the relationship between the topological and convergence closure product of two Fréchet spaces. In particular, we give a necessary and sufficient condition for the topological product of two compact Hausdorff Fréchet spaces to be a Fréchet space.     

Intersections of Fréchet Schwartz spaces and their duals

In this article the structure of the intersections of a Fréchet Schwartz space F and a (DFS)-space E=ind _n E _n is investigated. A complete characterization of the locally convex properties of E ⋃ F is given. This space is boraological if and only if the inductive limit E + F is complete. The results are based on recent progress on the structure of (LF)-spaces. The article includes examples of (FS)-spaces F and (DFS)-spaces E such that there are sequentially continuous linear forms on E ⋃ F which are not continuous, thus answering a question of Langenbruch. Acknowledgement: The results in this article were obtained during the author’s stay at the University of Paderborn, Germany, during the academic year 1994/95. The support of the Alexander von Humboldt Stiftung is greatly appreciated. The content of the article was presented as an invited paper in a Special Session of the AMS meeting in New York in April, 1996.     

Inverse mapping theorem in Fréchet spaces

Journal of Optimization Theory and Applications - We consider the classical inverse mapping theorem of Nash and Moser from the angle of some recent development by Ekeland and the authors....     

Characterizations of nuclearity in Fréchet spaces

We extend the result of A. Bellow (Proc. Nat. Acad. Sci. USA73, No. 6 (1976), 1798–1799) on the characterization of finite-dimensional Banach spaces, to a characterization of nuclearity for Fréchet spaces. Those spaces are nuclear iff every Pettis-bounded and Pettis-uniformly integrable amart is mean convergent. Several other characterizations are given.     

Weak and strong convergence of amarts in Fréchet spaces

Several new characterizations of nuclearity in Fréchet spaces are proved. The most important one states tat a Fréchet space is nuclear if and only if every mean bounded amart is strongly a.s. convergent. This extends the result in [A. Bellow, Proc. Nat. Acad. Sci. USA73, No. 6 (1976), 1798–1799] in a more positive way, and gives a different proof of it. The results of Brunel and Sucheston [C. R. Acad. Sci. Paris Ser. A (1976), 1011–1014], are extended to yield the same characterization of reflexivity of a Fréchet space in terms of weak convergence a.s. of weak amarts.     

Real analytic curves in Fréchet spaces and their duals

The following results are presented: 1) a characterization through the Liouville property of those Stein manifoldsU such that every germ of holomorphic functions on xU can be developed locally as a vector-valued Taylor series in the first variable with values inH(U); 2) ifT is a surjective convolution operator on the space of scalar-valued real analytic functions, one can find a solutionu of the equationT u=f which depends holomorphically on the parameterz wheneverf depends in the same manner. These results are obtained as an application of a thorough study of vector-valued real analytic maps by means of the modern functional analytic tools. In particular, we give a tensor product representation and a characterization of those Fréchet spaces or LB-spacesE for whichE-valued real analytic functions defined via composition with functionals and via suitably convergent Taylor series are the same.     

Resolutions of the identity in Fréchet spaces

It is a classical result that every Bade -complete Boolean algebra of (selfadjoint) projections in a separable Hilbert space coincides with the projections forming the resolution of the identity of some bounded selfadjoint operator. This result is extended to the setting of separable Fréchet spaces. Namely, every Bade -complete Boolean algebra of projections in such a space coincides with the resolution of the identity of some (continuous) scalar-type spectral operator having spectrum a compact subset of.     

Geometric properties of Fréchet spaces and selection of basis sequences

For an arbitrary nuclear Fréchet spaceE possessing the well-known geometric propertiesD ₁ (DN) and Ω, certain sequences of functionals and elements with power-type estimates of the norms are selected. With the help of these objects, two isomorphic spacesK ₁ andK ₂ of power series of infinite type and continuous mapsJ ₂: K₂→E andJ ₁: E→K₁ are defined. Under some auxiliary conditions, it is proved that the_i are isomorphisms and the spaceE possesses a basis. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 102–111, July, 1999.     

Fréchet algebras,formal power series,and analytic structure

The notion of locally Riemann algebras is introduced. By studying the ideal structure of Fréchet algebras, we provide sufficient conditions for the existence of local analytic structure in the spectrum of a Fréchet algebra, and, as a welcome bonus, we characterize locally Riemann algebras.     

A generalized vector-valued variational principle in Fréchet spaces

In the framework of Frechet spaces, we give a generalized vector-valued Ekeland＇s variational principle, where the perturbation involves the subadditive functions of countable generating semi-norms. By modifying and developing the method of Cammaroto and Chinni, we obtain a density theorem on extremal points of the vector-valued variational principle, which extends and improves the related known results.     

Block sequences in nuclear Fréchet spaces with basis

It is shown that a block sequence in a nuclear Fréchet space with a basis has a block extension if and only if the subspace it generates is complemented. In addition, a short proof is given of the following result of Dubinsky and Robinson: a nuclear Fréchet space is isomorphic to = R^N, N = {1,2,...} if it has a basis such that any block sequence with blocks of length 2 of any permutation of this basis has a block extension. It is shown that a similar result holds without considering permutations of the basis if the length of the blocks is arbitrary.Translated from Matematicheskie Zametki, Vol. 17, No. 6, pp. 899–908, June, 1975.The author wishes to thank B. S. Mityagin for calling his attention to this problem and for his valuable suggestions.     

On Ulm’s method for Fréchet differentiable operators

We provide new sufficient convergence conditions for the semilocal convergence of Ulm’s method (Izv. Akad. Nauk Est. SSR 16:403–411, 1967) in order to approximate a locally unique solution of an equation in a Banach space setting. We show that in some cases, our hypotheses hold true but the corresponding ones (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) do not. We also show that under the same hypotheses and computational cost as (Burmeister in Z. Angew. Math. Mech. 52:101–110, 1972; Kornstaedt in Aequ. Math. 13:21–45, 1975; Petzeltova in Comment. Math. Univ. Carol. 21:719–725, 1980; Potra and Ptǎk in Cas. Pest. Mat. 108:333–341, 1983; Ulm in Izv. Akad. Nauk Est. SSR 16:403–411, 1967) finer error sequences can be obtained. Numerical examples are also provided further validating the results.     

Sufficient sets in weighted Fréchet spaces of entire functions

Under study are sufficient sets in Fréchet spaces of entire functions with uniform weighted estimates. We obtain general results on the a priori overflow of these sets and introduce the concept of their minimality. We also establish necessary and sufficient conditions for a sequence of points on the complex plane to be a minimal sufficient set for a weighted Fréchet space. Applications are given to the problem of representation of holomorphic functions in a convex domain with certain growth near the boundary by exponential series.     

Non-quasianalytic curves in Fr��chet spaces

It is proved that every weakly non-quasianalytic ultradifferentiable curve with values in a Fréchet space E is topologically (or strongly) ultradifferentiable if and only if the space E satisfies the topological invariant (DN), thus solving a problem posed by Kriegl and Michor.