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We characterize the condition (Ω) for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions (P Ω) and (P Ω ¯ ¯) for distributional kernels are characterized in a similar way. By lifting theorems for Fréchet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space { f ∈ E (X) | P (D) f = 0 } satisfies (Ω) for any differential operator P(D) and any open convex set X ⊆ R d . [ABSTRACT FROM AUTHOR]

We study an extension problem for continuous linear maps in the setting of (LB)-spaces. More precisely, we characterize the pairs (E, Z), where E is a locally complete space with a fundamental sequence of bounded sets and Z is an (LB)-space, such that for every exact sequence of (LB)-spaces the map L (Y , E) → L (X , E) , T ↦ T ∘ ι is surjective, meaning that each continuous linear map X → E can be extended to a continuous linear map Y → E via ι , under some mild conditions on E or Z (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24]. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the question of surjectivity of the Cauchy-Riemann operator ∂ ¯ on spaces E V (Ω , E) of C ∞ -smooth vector-valued functions whose growth on strips along the real axis with holes K is induced by a family of continuous weights V . Vector-valued means that these functions have values in a locally convex Hausdorff space E over C . We derive a counterpart of the Grothendieck-Köthe-Silva duality O (C \ K) / O (C) ≅ A (K) with non-empty compact K ⊂ R for weighted holomorphic functions. We use this duality and splitting theory to prove the surjectivity of ∂ ¯ : E V (Ω , E) → E V (Ω , E) for certain E. This solves the smooth (holomorphic, distributional) parameter dependence problem for the Cauchy-Riemann operator on E V (Ω , C) . [ABSTRACT FROM AUTHOR]

We study weighted (PLB)-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck's classical result that the space O M of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not. [ABSTRACT FROM AUTHOR]

We give a complete solution to the Borel–Ritt problem in non-uniform spaces A (M) - (S) of ultraholomorphic functions of Beurling type, where S is an unbounded sector of the Riemann surface of the logarithm and M is a strongly regular weight sequence. Namely, we characterize the surjectivity and the existence of a continuous linear right inverse of the asymptotic Borel map on A (M) - (S) in terms of the aperture of the sector S and the weight sequence M. Our work improves previous results by Thilliez (Results Math 44:169–188, 2003) and Schmets and Valdivia (Stud Math 143:221–250, 2000). [ABSTRACT FROM AUTHOR]

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 ∞ (K) = { F | K : F ∈ C ∞ (R) } to C ∞ (R) . This allows us to deal with examples of the form K = { a n : n ∈ N } ∪ { 0 } for a n → 0 previously considered by Fefferman and Ricci as well as Vogt. [ABSTRACT FROM AUTHOR]

We show how to define invariants of graphs related to quantum sl 2 when the graph has more then one connected component and components are colored by blocks of representations with zero quantum dimensions.

In this paper, we examine certain cardinal properties of subspaces N τ φ X of the space NX of complete linked systems of a topological space X. [ABSTRACT FROM AUTHOR]

In this paper, we examine categorical and cardinal properties of hyperspaces with finite number of components. We prove that the functor Cn : Comp → Comp is not normal, i.e., it does not preserve epimorphisms of continuous mappings. We also discuss the density, the caliber, and the Shanin number of the space Cn(X). [ABSTRACT FROM AUTHOR]

In this paper, the edge version of SDD and ISI index are introduced. Using this concept the explicit formulas for the edge version of SDD and ISI of rooted product graphs and i-th vertex rooted product graphs are reckoned. [ABSTRACT FROM AUTHOR]

We provide a sufficient condition for a linear differential operator with constant coefficients P(D) to be surjective on C∞(X) and D′(X), respectively, where X⊆Rd is open. Moreover, for certain differential operators this sufficient condition is also necessary and thus a characterization of surjectivity for such differential operators on C∞(X), resp. on D′(X), is derived. Additionally, we obtain for certain surjective differential operators P(D) on C∞(X), resp. D′(X), that the spaces of zero solutions CP∞(X)={u∈C∞(X);P(D)u=0}, resp. DP′(X)={u∈D′(X);P(D)u=0} possess the linear topological invariant (Ω) introduced by Vogt and Wagner (Stud. Math. 68:225-240, 1980), resp. its generalization (PΩ) introduced by Bonet and Domański (J. Funct. Anal. 230:329-381, 2006). [ABSTRACT FROM AUTHOR]

In the work we investigate categorical and topological properties of the functor O S τ of semiadditive τ -smooth functionals in the category T y c h of Tychonoff spaces and their continuous mappings, which extends the functor OS of semiadditive functionals in the category C o m p of compact spaces and their continuous mappings. It is proved that the functor O S τ is a normal functor in the category T y c h of Tychonoff spaces. [ABSTRACT FROM AUTHOR]

This paper deals with the noncommutative Fréchet [ABSTRACT FROM AUTHOR]

We consider the Fréchet $${}^*$$ -algebra $${\mathcal {L}}(s',s)\subseteq \mathcal L(\ell _2)$$ of the so-called smooth operators, i.e. continuous linear operators from the dual $$s'$$ of the space s of rapidly decreasing sequences to s. This algebra is a non-commutative analogue of the algebra s. We characterize closed $${}^*$$ -subalgebras of $${\mathcal {L}}(s',s)$$ which are at the same time isomorphic to closed $${}^*$$ -subalgebras of s and we provide an example of a closed commutative $${}^*$$ -subalgebra of $${\mathcal {L}}(s',s)$$ which cannot be embedded into s. [ABSTRACT FROM AUTHOR]

The aim of paper is to give some results, that prepare for studying the problem on cross theorems for separately $$(\cdot , W)$$ -meromorphic functions. Some general versions of extension theorem of Levi type are extended to the classes of meromorphic functions f on $$D \times (\Delta _r {\setminus } \overline{\Delta })$$ with values in a locally convex space F. Here, the function f is assumed that, for each $$z \in D_*,$$ the function $$f_z = f(z, \cdot )$$ has a ( F, W)-meromorphic extension to $$\Delta _r,$$ where F is either a locally (or sequentially) complete locally convex space or a Fréchet space, the space $$W \subseteq F'$$ is separating (or determines boundedness), $$\Delta _r = \{\lambda \in {\mathbb C}: |\lambda | < r\}$$ with $$r > 1, \Delta = \Delta _1$$ and D is either a domain in $${\mathbb C}^n$$ or a balanced domain in a Fréchet space containing a non-pluripolar balanced convex compact subset, $$D_*$$ is dense in D. [ABSTRACT FROM AUTHOR]

The aim of this paper is to find some necessary and sufficient conditions by means of the property ( DN) of E and the property ( LB) of F under which H( X, F) = H( X, F) and H( X, H( F*)) = H( X, H( F*)) where E, F are Fréchet spaces and X is a compact determining polydisc in E*. At the same time, we also study the problem of the local Dirichlet representation of separately holomorphic functions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some criterions for holomorphy of F-valued σ( F, W)-holomorphic functions which are bounded on bounded sets in a domain D of Fréchet spaces E (resp. $${\mathbb{C}^n}$$) where $${W \subset F'}$$ defines the topology of Fréchet space F. Base on these results we consider the problem on holomorphic extension of F-valued σ( F, W)-holomorphic functions from non-rare subsets of D and from subsets of D which determines uniform convergence in H( D). As an application of the above, some theorems of Vitali-type for a locally bounded sequence $${\{f_i\}_{i \in \mathbb{N}}}$$ of Fréchet-valued holomorphic functions are also proved. [ABSTRACT FROM AUTHOR]

The aim of this paper is to obtain a version of continuous functional calculus and some new envelope representation results in vector lattices as well as to indicate some applications. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We relate the equivalence of the topologies τ o and τ ω on a fully nuclear space E having the bounded approximation property with the polynomially significant properties on E ′ b , using the localization property of Defant and Govaerts (A. Defant and W. Govaerts (1986). Tensor products and spaces of vector-valued continuous functions. Manuscripta Math. , 55 , 433-449). This allows us to give examples of Fréchet nuclear spaces with bases E and F so that τ o =τ ω on . We also give an example of Fréchet nuclear spaces with bases E and F so that τ o =τ ω on for every open polydisc U in E × F ′ ] . The conditions for equivalence of topologies are expressed in terms of the linear invariants (DN), and given in Vogt (D. Vogt (1983). Frécheträume, zwischen denen jede stetige lineare Abbildung beschränkt ist. J. reine u. angew. Math. , 345 , 182-200.) and Meise and Vogt (R. Meise and D. Vogt (1986). Holomorphic functions of uniformly bounded type on nuclear Fréchet spaces. Studia Math. , 83 , 147-175.). [ABSTRACT FROM AUTHOR]