Pawel Domanski

• Prof. dr hab.
• Professor (Full) at Adam Mickiewicz University in Poznań

The presentation of Equation was incorrect in the proof section of the Theorem 2.18.

We obtain a complete characterization of surjective Hadamard type operators \(H_T,T\in C^\infty (\mathbb {R}^d)'\) (i.e. of multiplicative convolution operators) on \(C^\infty (\mathbb {R}^d)\) using a restrictive slowly decreasing condition and a division property both new and valid for the Mellin transform \(\mathscr {M}(T)\). We also characteriz...

In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.

In this paper the spectrum of composition operators on the space of real analytic functions is investigated. In some cases it is completely determined while in some other cases it is only estimated.

We describe all Euler partial differential operators which act on the space of real analytic functions and we identify them among the Taylor multipliers on these spaces. Partial differential operators of the form, where qj,1,qj,0:(aj,bj)→C, are called generalized Euler differential operators whenever all Dj are conjugate to the classical Euler diff...

We prove criteria for global analytic solvability for some Euler type partial differential equations — a class of linear partial differential equations with variable (polynomial) coefficients. This is based on a very general Mellin type theory developed in the present paper which is valid for analytic functionals (in particular, distributions of co...

We consider multipliers on the space of real analytic functions of several variables A(Ω), Ω⊂Rd open, i.e., linear continuous operators for which all monomials are eigenvectors. If zero belongs to Ω these operators are just multipliers on the sequences of Taylor coefficients at zero. In particular, Euler differential operators of arbitrary order ar...

We obtain full description of eigenvalues and eigenvectors of composition operators \({C_{\varphi}:\fancyscript{A}\mathbb{R}\to \fancyscript{A}\mathbb{R}}\) for a real analytic self map \({\varphi:\mathbb{R} \rightarrow \mathbb{R} }\) as well as an isomorphic description of corresponding eigenspaces. We completely characterize those \({\varphi}\) f...

We consider multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We characterize sequences of complex numbers which are sequences of eigenvalues for some multiplier. We characterize invertible multipliers, in particular, we find which Euler differential operators of infinite order have g...

We provide a complete solution of the abstract Cauchy problem for operator valued Laplace distributions or hyperfunctions on complete ultrabornological locally convex spaces (like spaces of smooth functions and distributions). This extends results of Komatsu for operators on Banach spaces. Concrete examples are provided. The crucial tools for our s...

We consider multipliers on the spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We prove representation theorems in terms of analytic functionals and in terms of holomorphic functions. We characterize Euler differential operators among multipliers. Then we characterize when such operators are surje...

We study the dynamical behaviour of composition operators C ϕ defined on spaces A (Ω) of real analytic functions on an open subset Ω of R d . We characterize when such operators are topologically transitive, i.e. when for every pair of non-empty open sets there is an orbit intersecting both of them. Moreover, under mild assumptions on the compositi...

We consider the topological algebra of (Taylor) multipliers on spaces of real analytic functions of one variable, i.e., maps for which monomials are eigenvectors. We describe multiplicative functionals and algebra homomorphisms on that algebra as well as idempotents in it. We show that it is never a Q-algebra and never locally mconvex. In particula...

We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an example of a semi-proper map ' such that the associated composition operator is not open onto its image.We characterize composition operators on spaces of real analytic functions which are open onto their images. We give an exampl...

We characterize those composition operators defined on spaces of holomorphic functions of several variables which are power bounded, i.e. the orbits of all the elements are bounded. This condition is equivalent to the composition operator being mean ergodic. We also describe the form of the symbol when the composition operator is mean ergodic. Key...

We study the dynamical behaviour of composition operators defined on spaces of real analytic functions. We characterize when such operators are power bounded, i.e. when the orbits of all the elements are bounded. In this case this condition is equivalent to the composition operator being mean ergodic. In particular, we show that the composition ope...

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f_\lambda)\subseteq \mathscr_{\{\omega\}}(\Omega)$ of ultradifferentiable functions of Roumieu type (in particular, of real analytic functions or of functions from Gevrey classes)...

We introduce a Laplace transform for Laplace hyperfunctions valued in a complete locally convex space $X$. In this general case the Laplace transform is a compatible family of holomorphic functions with values in local Banach spaces. Especially interesting is the case where $X=L_b(E,F)$ is the space of operators between locally convex spaces. In th...

We characterize when an ideal of the algebra A(\mathbbRd){A(\mathbb{R}^d)} of real analytic functions on \mathbbRd{\mathbb{R}^d} which is determined by the germ at \mathbb Rd{\mathbb {R}^d} of a complex analytic set V is complemented under the assumption that either V is homogeneous or VÇ\mathbbRd{V\cap \mathbb{R}^d} is compact. The charact...

We consider the problem of real analytic parameter dependence of solutions of the linear partial differential equation $P(D)u=f$, i.e., the question if for every family $(f\sb\lambda)\subseteq \mathscr{D}'(\Omega)$ of distributions depending in a real analytic way on $\lambda\in U$, $U$ a real analytic manifold, there is a family of solutions $(u\s...

We study those Kothe co-echelon sequence spaces kp(V );1 • p • 1 or p = 0, which are locally convex (Riesz) algebras for the pointwise multiplication. We characterize in terms of the matrix V = (vn)n when an algebra kp(V ) is unital, locally m-convex, a Q-algebra, it has a continuous (quasi)-inverse, all entire functions act on it or some transcend...

We develop the theory of hyperfunctions with values in a locally convex non-necessarily metrizable space E and necessary conditions and sufficient conditions such that a rea- sonable theory of E-valued hyperfunctions exists. In particular, we show that it exists for various spaces of distributions but there is no such theory for the spaces of real...

We investigate the splitting of short exact sequences of the form0→X→Y→E→0, where E is the dual of a Fréchet Schwartz space and X, Y are PLS-spaces, like the spaces of distributions or real analytic functions or their subspaces. In particular, we characterize pairs (E,X) as above such that Ext1(E,X)=0 in the category of PLS-spaces and apply this ch...

The paper contains a comment and a review on a paper of Stefan Banach from 1939 devoted to his version of "Loi Supreme" of Hoene-Wroński and it tries to explain the content of the paper on the background and in terms of modern functional analysis.

It is shown that spaces of quasianalytic ultradifferentiable functions of Roumieu type ℰ{w}(Ω), on an open convex set (W) Í \mathbbRd(\Omega)\,{\subseteq}\,{\mathbb{R}}^d, satisfy some new (Ω) -type linear topological invariants. Some consequences for the splitting of short exact sequences of these spaces as well as for the structure of the space...

Based on the methods from interpolation theory we give a characterization of pairs (E, F) of Fréchet-Hilbert spaces so that for each Fréchet–Hilbert space G each short exact sequence 0 ® F ® G ® E ® 00 \rightarrow F \rightarrow G \rightarrow E \rightarrow 0 splits. This characterization essentially depends on a key condition (S) of an interpolation...

We characterize composition operators on spaces of real analytic functions which at the same time have closed image and are open onto their images. Under some mild assumptions, we also characterize composition operators with closed range and composition operators open onto their images. 2000 Mathematics Subject Classification 46E10, 47B33, 32C07 (p...

We show that a linear partial differential operator with constant coefficients P(D) is surjective on the space of E-valued (ultra-)distributions over an arbitrary convex set if E′ is a nuclear Fréchet space with property (DN). In particular, this holds if E is isomorphic to the space of tempered distributions S′ or to the space of germs of holomorp...

We give some necessary and some sufficient conditions for composition operators on spaces of real analytic functions to have closed range and to be open onto their ranges. We also prove sequential density of polynomials in the space of real analytic functions over an arbitrary compact subset of ℝ d.

We study the problem when an infinite system of linear functional equations mu(n)(f) = b(n) for n is an element ofN has a real analytic solution f on omega subset of or equal to R-d for every right-hand side (b(n))(nis an element ofN) subset of or equal to C and give a complete characterization of such sequences of analytic functionals (mu(n)). We...

This paper is an extended version of an invited talk presented during the Orlicz Centenary Conference (Poznań, 2003). It contains a brief survey of applications to classical problems of analysis of the theory of the so-called PLS-spaces (in particular, spaces of distributions and real analytic functions). Sequential representations of the spaces an...

We consider the complex 0 → Λ0(M; E) ∂ ω→ Λ1(M; E) ∂ ω→ ⋯ ∂ω→ Λ m(M; E), where E is a finite-dimensional vector bundle over a suitable differential manifold M, Λq(M; E) denotes the space of all smooth or real analytic or holomorphic sections of the q-exterior product of E and ∂ω(η) := ω Λ η for ω ∈ λ1(M; E). We give sufficient and necessary conditi...

We prove that the space of real analytic functions A(W){\cal A}(\Omega) on an arbitrary open set W Í \mathbbRd\Omega \subseteq \mathbb{R}^d has a Frchet infinite dimensional quotient space with a continuous norm.

Let Ω1, Ω2 be open subsets of ℝ and ℝ , respectively, and let A(Ω1) denote the space of real analytic functions on Ω1. We prove a Glaeser type theorem by characterizing when a composition operator Cφ : A(Ω1) → A(Ω2), Cφ(f) ≔ f ∘ φ, is a topological embedding. Using this result we characterize when A(Ω1) can be embedded topologically into A(Ω2) as a...

We give new characterizations of the subsets S of the unit disc of the complex plane such that the topology of the space A−∞ of holomorphic functions of polynomial growth on coincides with the topology of space of the restrictions of the functions to the set S. These sets are called weakly sufficient sets for A−∞. Our approach is based on a study o...

We characterize all Fréchet quotients of the space A(Ω) of (complex-valued) real-analytic functions on an arbitrary open set Ω ⊆ ℝ d. We also characterize those Fréchet spaces E such that every short exact sequence of the form 0 → E → X → A(Ω) → 0 splits.

Ideals of extendable and liftable operators are introduced giving a new approach to the study of the splitting of short exact sequences of Banach spaces. Maximality, duality and closedness with respect to pointwise bounded limits of the ideals are considered. Several examples are summarized and the role of L1-and L∞-spaces is clarified. Ideales de...

Let $\omega \subseteq {\bb R}^d$ be an open domain. The sequentially complete DF-spaces $E$ are characterized such that for each (some) discrete sequence $(z_n) \subseteq \omega$ , a sequence of natural numbers $(k_n)$ and any family $(x_{n, \alpha})_{n \in {\bb N}, \vert \alpha\vert \leqslant k_n} \subseteq E$ the infinite system of equa...

We give an elementary approach which allows us to evaluate Seip's conditions characterizing interpolating and sampling sequences in weighted Bergman spaces of infinite order for a wide class of weights depending on the distance to the boundary of the domain. Our results give also some information on cases not covered by Seip's theory. Besides, we a...

The paper gives a survey of the linear topological properties of the space of analytic func- tions on the real line. Besides the analysis of the classical properties, the emphasis is put on certain, very useful “splitting lemma1. The proofs which are presented here are based on much more elementary tools than the known proofs for the same space on...

Let X be a Banach space. It is proved that the composition operator on X -valued Hardy spaces, weighted Bergman spaces and Bloch spaces is weakly compact or Rosenthal if and only if both id: X → X and the corresponding composition operator on scalar valued spaces are weakly compact or Rosenthal, respectively.

Let Ω ⊆ ℝn be an open set and let A(Ω) denote the class of real analytic functions on Ω. It is proved that for every surjective linear partial differential operator P(D,x) : A(Ω) → A(Ω) and every family (fλ) ⊆ A(Ω) depending holomorphically on λ ∈ ℂm there is a solution family (uλ) ⊆ A(Ω) depending on λ in the same way such that P(D,x)uλ = fλ, for...

Let Ω ⊆ R n \Omega \subseteq \mathbb {R}^n be an open set and let A ( Ω ) \mathrm {A}(\Omega ) denote the class of real analytic functions on Ω \Omega . It is proved that for every surjective linear partial differential operator P ( D , x ) : A ( Ω ) → A ( Ω ) P(D,x):\mathrm {A}(\Omega )\to \mathrm {A}(\Omega ) and every family ( f λ ) ⊆ A ( Ω ) (f...

It is proved that if Ti are linear continuous operators then the following (algebraically) exact complex: (formula presented) splits for i ≧ 1, i.e., Ti : (script′(Ωi))si → Im Ti has a continuous and linear right inverse. If T0 is a matrix of convolution operators, then the complex (*) splits for i = 0 if and only if F is a strict projective limit...

Let Ω be an open connected subset of ℝd. We show that the space A(Ω) of real-analytic functions on Ω has no (Schauder) basis. One of the crucial steps is to show that all metrizable complemented subspaces of A(Ω) are finite-dimensional.

The splitting problem is studied for short exact sequences consisting of countable projective limits of DFN-spaces (*) 0 → F → X → G → 0, where F or G are isomorphic to the space of distributions D′. It is proved that every sequence (*) splits for F ≃ D′ iff G is a subspace of D′ and that, for ultrabornological F, every sequence (*) splits for G ≃...

Every weakly compact composition operator between weighted Banach spaces H∞v of analytic functions with weighted sup-norms is compact. Lower and upper estimates of the essential norm of continuous composition operators are obtained. The norms of the point evaluation functionals on the Banach space H∞v are also estimated, thus permitting to get new...

For a wide class of weights we find the approximative point spectrum and the essential spectrum of the pointwise multiplication operator Mφ, Mφ(f) = φf, on the weighted Banach spaces of analytic functions on the disc with the sup-norm. Thus we characterize when Mφ is Fredholm or is an into isomorphism. We also study cyclic phenomena for the adjoint...

We prove the following common generalization of Maurey's extension theorem and Vogt's (DN)-(W) splitting theorem for Fréchet spaces: if T is an operator from a subspace E of a Fréchet space G of type 2 to a Fréchet space F of dual type 2, then T extends to a map from G into F'' whenever G/E satisfies (DN) and F satisfies (W).

An example of two distinguished Frechet spaces E, F is given (even more, E is quasinormable and F is normable) such that their completed injective tensor product E (X) over cap(epsilon)F is not distinguished. On the other hand, it is proved that for arbitrary reflexive Frechet space E and ;arbitrary compact set K the space of E-valued continuous fu...

We will show using purely linear functional analytic methods that each exact complex0→F→(C∞(Ω))s0→p0(C∞(Ω))s1→p1(C∞(Ω))s2→p2…,wherepiare matrices of convolution operators (in particular, linear differential operators with constant coefficients), splits fromp1on (in the category of topological vector spaces). Moreover, we characterize when these com...

We characterize those analytic self-maps of the unit disc which generate bounded or compact composition operators C between given weighted Banach spaces H∞ v or H0 v of analytic functions with the weighted sup-norms. We characterize also those composition operators which are bounded or compact with respect to all reasonable weights v.

It is proved that if a Köthe space λ1(A) is distinguished and E is an arbitrary Fréchet space then every reflexive map T: λ1(A)→E (i.e., T maps bounded sets into relatively weakly compact ones) factorizes through a reflexive Fréchet space. An analogous result is proved for Montel maps (i.e., which map bounded sets into relatively compact ones). The...

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 ana...

It is proved that for any coechelon space kiV) of order p, 1 ~ p ~ 00, and any compact set K, the space of conti nuous functions C(K, kiV)) is bornological. This is a par­ tial solution of the problem of Schmets and Bierstedt on bornologicity of LB-spaces of continuous functions. More­ over, if kpCV) is Montel, then C(K, kpCV)) is even the local co...

We prove that for Banach spaces E,F,G,H and operators T∈ℒ(E,G), S∈ℒ(F,H) the tensor product T⊗S:E⊗ ε F→G⊗ ε H is a Grothendieck operator, provided T is a Grothendieck operator and S is compact.

We prove that for any Köthe matrices a and b if T : λ1(α) → λ0(b) maps bounded sets into relatively compact sets, then T factorizes through a Fréchet Montel space. This is a consequence of a given description of those compact subsets in a coechelon space k∞(v) of type oo which are contained in an absolutely convex hull of a null sequence. An exampl...

We will show that for each sequence of quasinormable Fréchet spaces (E n ) n∈ℕ there is a Köthe space λ(A) such that Ext 1 (λ(A),λ(A))=Ext 1 (λ(A),E n )=0 and there are exact sequences of the form ⋯→λ(A)→λ(A)→λ(A)→λ(A)→E n →0· If, for a fixed n∈ℕ, E n is nuclear or a Köthe sequence space, the resolution above may be reduced to a short exact sequenc...

Let E and F be Fréchet spaces. We prove that if E is reflexive, then the strong bidual (E⊗ ^ ε F) b '' is a topological subspace of L b (E b ' ,F b '' ). We also prove that if, moreover, E is Montel and F has the Grothendieck property, then E⊗ ^ ε F has the Grothendieck property whenever either E or F b '' has the approximation property. A similar...

Let E, F be Banach spaces. If F is of finite cotype p and there is a non-compact (non-limited) map T : l p → F , then we characterize those E for which K(E, F) contains a (complemented) copy of c 0. If either E or F has an unconditional basis and there is a non-limited operator T : E → F , then K(E, F) contains a complemented copy of c 0. Analogous...

Let E, F be either Fréchet or complete DF-spaces and let A(E, F) ⊆ B(E, F) be spaces of operators. Under some quite general assumptions we show that: (i) A(E, F) contains a copy of c 0 if and only if it contains a copy of l ∞; (ii) if c 0 ⊆ A(E, F), then A(E, F) is complemented in B(E, F) if and only if A(E, F) = B(E, F); (iii) if E or F has an unc...

Let K be a non-empty compact subset of a Fréchet space E and let X be a Banach space. By means of a given representation of the LB-space H(K, X) of germs of holomorphic functions with values in X as a space of linear operators, it is proved that the space H(K, X) is complete if E is quasinormable or if X is complemented in its bidual. If E is a Fré...

Consider the following conditions: (a) Every regular LB-space is complete; (b) if an operator T between complete LB-spaces maps bounded sets into relatively compact sets, then T factorizes through a Montel LB- space; (c) for every complete LB-space E the space C(βℕ,E) is bornological. We show that (a)⇒(b)⇒(c). Moreover, we show that if E is Montel,...

We prove that the space of Riemann integrable functions is uncomple- mented in L∞(0; 1) since it contains a complemented copy of c0.

This paper is related to the authors' recent articles. In particualr, it is shown that the Banach space of vector-valued continuous functions is usually uncomplemented in the larger spaces of functions continuous with respect to some weaker topologies in the range Banach space. On the other hand, it is proved in earlier papers of the same authors t...

We sovle in the negative a problem of Wolfe ifC(T A ) is an injective Banach space wheneverC(T) is injective,T compact, andT A is the Amir boundary ofT (i.e., the complement of the maximal open extremally disconnected subset ofT). In particular, we findT such thatC(T) is aP 3-space andT A ∼βN\N.

We prove some topological properties of completely regular (esp. locally compact) topological spaces T such that the locally convex (esp. Fréchet) space C(T) equipped with the compact-open topology is injective (i.e., complemented in every locally convex space containing it isomorphically). Examples are presented which show that the results obtaine...

It is proved that every complemented subspace of an arbitrary topological product of (nonnecessarily separable) Hubert spaces is isomorphic to a product of Hubert spaces. A counterexample is given showing that this result cannot be proved by the same direct method as for countable products.

It is proved that every complemented subspace of an arbitrary topological product of (nonnecessarily separable) Hilbert spaces is isomorphic to a product of Hilbert spaces. A counterexample is given showing that this result cannot be proved by the same direct method as for countable products.

Introduction. The main aim of this note is to prove the useful principles of local reflexivity for linear operators and for locally convex (esp. Fr6chet) quojections. We also obtain a special dual form of the principle for strict LB-spaces. We start with the following natural result which is an operator form of the principle.

It is proved that: (i) every complemented subspace in an infinite product of L1-predual Banach spaces $\prod_{i\in I} X_i$ is isomorphic to Z × Km, where $\dim K = 1, \mathfrak{m} \leq \operatorname{card}I$ and Z is isomorphic to a complemented subspace of $\prod_{i\in J} X_i, J \subseteq I, Z$ contains an isomorphic copy of $c_O^{\operatorname{car...

A twisted sum of (topological vector) spaces Y and Z is a space X with a subspace V, isomorphic to Y1 for which X/Y1 is isomorphic to Z. It splits if Y1 is complemented. It is proved that every twisted sum of a Banach space Y and a nuclear space Z splits. Köthe sequence spaces Z for which this holds are characterized. Every locally convex twisted s...