Reinhold Meise

Köthe Sets and Köthe Sequence Spaces

Publisher Summary This chapter discusses the Kothe sets and Kothe sequence spaces. Echelon and co-echelon spaces had been studied by G. Kothe (and O. Toeplitz) prior to the development of general tools available through the present day theory of topological vector spaces; Kothes early work with sequence spaces has helped point the way in establishing a general theory. In its turn, however, this general theory has been successfully utilized in the study of sequence spaces, while echelon and co-echelon spaces have continued to serve as a ready source for examples and counter examples. The chapter presents the fundamental definitions and establish the notation and treats the role of the space K p in the duality of echelon and co-echelon spaces. A condition on the Kothe matrix A is considered and is preferred to phrase in terms of the corresponding decreasing sequence—namely, the sequence space analog of the property that is called “regularly decreasing.”

Phragmén-Lindelöf principles on algebraic varieties

From several results in recent years, starting with H6rmanders characterization of the constant coefficient partial differential equations P(D)u = f that have a real analytic solution u for every real analytic function f, it has become clear that certain properties of the partial differential operator P(D) are equivalent to estimates of Phragmen-Lindelof type for plurisubharmonic functions on the algebraic variety

Distinguished Echelon Spaces and the Projective Description of Weighted Inductive Limits of Type (X)

Publisher Summary This chapter discusses the distinguished echelon spaces and provides a projective description of weighted inductive limits of type ұdҞ(X). The general problem of a projective description for weighted inductive limits of spaces of continuous and holomorphic functions was raised. This problem remains open, but a new, very weak, condition sufficient for distinguishedness of echelon spaces is presented in the chapter. The property generalizes the quasi-normable and the reflexive (or, equivalently, Montel) cases of distinguished echelon spaces λ1. The chapter proves that even the weaker hypothesis is sufficient to imply that λ1 is distinguished. The chapter also demonstrates that the arguments used in the sequence space setting can be modified to yield a similar, rather general, result on the topological equality.

The geometry of analytic varieties satisfying the local Phragmén-Lindelöf condition and a geometric characterization of the partial differential operators that are surjective on {}({ℝ}⁴)

The local Phragmen-Lindelof condition for analytic subvarieties of C n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hormander has shown. Here, necessary geometric conditions for this Phragmen-Lindelof condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in C 3 . The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on R 4 .

Applications of the Projective Limit Functor to Convolution and Partial Differential Equations

Let e {ω{(ℝ N ) denote the non-quasianalytic class of all {ω{-ultradifferentiable functions on ℝ N . This notion is an extension of the classical Gevrey classes Γ{d{(ℝ N ), d > 1. Reporting on our work [5] and [6], we explain how the projective limit functor introduced by Palamodov [21] can be used to characterize the surjectivity of (1) convolution operators T μ on e {ω{(ℝ) and (2) linear partial differential operators P(D) on e {ω{(ℝ N ).

Splitting of closed ideals in ()-algebras of entire functions and the property ()

For a plurisubharmonic weight function p on cn let Ap(Cn) denote the (DFN)-algebra of all entire functions on cn which do not grow faster than a power of exp(p). We prove that the splitting of many finitely generated closed ideals of a certain type in Ap(Cn), the splitting of a weighted 0-complex related with p, and the linear topological invariant (DN) of the strong dual of Ap(Cn) are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of p, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions p, this characterization reduces to a covexity property of the inverse of p. Using these criteria, we present a wide range of examples of weights p for which the equivalences stated above hold and also where they fail. For p a nonnegative plurisubharmonic (psh) function on cn) let Ap(Cn) denote the algebra of all entire functions f such that If(Z)l 0 depending on f. Algebras of this type arise at various places in complex analysis and functional analysis, e.g. as Fourier transforms of certain convolution algebras. The structure of their closed ideals has been studied for a long time, primarily in the work of Schwartz [25], Ehrenpreis [9], Malgrange [17], and Palamodov [23] in connection with the existence and approximation of (systems of) convolution equations. The question whether a certain parameter dependence of the right-hand side of such an equation is shared also by its solutions, is closely related with the question of the existence of a continuous linear right inverse. The existence of such a right inverse is equivalent to the splitting of the closed ideal I associated to the corresponding equation. Also, since the quotient space Ap(Cn)/I is quite often identified with the space Ap(V) of holomorphic functions on the variety V of I which satisfy the restricted growth conditions, the latter question is equivalent to the existence of a linear extension operator from Ap(V) to Ap(Cn). Answers to these questions for various algebras have been given e.g. by Grothendieck (see lVeves [28]), Cohoon [7], Djakov and Mityagin [8], and Vogt [33]. The fact that for P(z) = IZl8, s > 1 all closed ideals in Ap(C) are cornplemented, was observed by Taylor [27]. Then Meise [19] extended Taylors resultsX using a more functional analytic approach. He showed that the structural property (DN) of the strong dual Ap(Cn)b of Ap(Cn) implies that all slowly decreasing ideals Received by the editors July 29, 1986. 1980 Mathematics Subject Classification (1985 Reon). Primary 32E25, 46E25; Secondary 46A12, 32Al .

Lokalkonvexe Unterräume in Topologischen Vektorräumen und Das ε-Produkt

It is well-known that the algebraic tensor product E ⊗ Y of a not necessarily locally convex topological vector space E and a locally convex space Y can be identified with a subspace of the so-called ε-product EεY (a space of continuous linear mappings from Y′ into E). So, whenever EεY is complete, even the completed tensor product is (isomorphic to) a subspace of EεY. As this occurs in many important cases, it is interesting to remark that, for each continuous linear operator u from a locally convex space F into E, there exists a locally convex U with continuous embedding j∶U→E and a continuous linear map û∶F→U such that u=j·û. As main applications of a combination of these ideas, we obtain a characterization of the functions in as continuous functions with values in locally convex spaces (this gives new aspects for the intergration theory of Gramsch [5]) and a result extending a theorem in [6] on holomorphic functions with values in non locally convex spaces to arbitrary complex manifolds.

On the range of the Borel map for classes of non-quasianalytic functions

Let e(ω) (IRN) denote the non-quasianalytic class of ω-ultradifferentiable functions of Beurling type on IRN. Using a characterization of the bornological linear subspaces of (DFS)-spaces, an idea of Carleson and Hormanders solution of the -problem, it is characterized for which weight functions σ a certain Kothe sequence space Λ(σ, N) is contained in the range of the Borel map BN: f → (f(α)(0))α∈IX0N. The same question is treated also for the class σ(ω)(IRN) of ω-ultradifferentiable functions of Roumieu type.

Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces

SummaryLetP be a complex polynomial inn variables of degree 2 andP(D) the corresponding partial differential operator with constant coefficients. It is shown thatP (D) :C∞(ℝn) →C∞(ℝn) admits a continuous linear right inverse if and only if after a separation of variables and up to a complex factor for some c ∈ ℂ the polynomialP has the form