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We consider weighted Bergman spaces $$A_\mu ^1$$ A μ 1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of $$A_\mu ^1$$ A μ 1 . Also, as a consequence of a characterization of solid $$A_\mu ^1$$ A μ 1 -spaces, we show that, in the case of entire functions, there indeed exist solid $$A_\mu ^1$$ A μ 1 -spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of $$A_\mu ^1$$ A μ 1 , when $$\mu$$ μ equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted $$A^1$$ A 1 - and $$H^\infty$$ H ∞ -spaces, the validity of which requires the assumption that $$-\log v$$ - log v belongs to the class $$\mathcal {W}_0$$ W 0 , studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight $$v(z) = \exp ( -1 /(1-|z|)$$ v ( z ) = exp ( - 1 / ( 1 - | z | ) provides an example satisfying both assumptions.

[EN] We consider the Cauchy problem for a general class of parabolic partial differential equations in the Euclidean space R-N. We show that given a weighted L-p-space L-w(p)(R-N) with 1 0 such that, if the initial data f belongs to the closed linear span of e(n) with n >= n(m), then the decay rate of the solution of the problem is at least t(-m) for large times t. The result generalizes the recent study of the authors concerning the classical linear heat equation. We present variants of the result having different methods of proofs and also consider finite polynomial decay rates instead of unlimited m., The research of Bonet was partially supported by the projects MTM2016-76647P and GV Prometeo/2017/102 (Spain). The research of Taskinen was partially supported by a research grant from the Faculty of Science of the University of Helsinki.

[EN] There are recent results concerning the boundedness and also unboundedness of Bergman projections on weighted spaces of the unit disc in special cases of rapidly decreasing weights, i.e. "large" Bergman spaces. The aim of our paper is to show that the cases of boundedness are largely exceptional: in general the Bergman projections are unbounded. In addition we give a new, more functional analytic proof for the known central boundedness case which also enables us to transfer our results to harmonic Bergman spaces., The research of Bonet was partially supported by the project MCIN PID2020-119457GB-I00/AEI/10.13039/501100011033. The research of Taskinen was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

[EN] We determine the solid hull for 2 < p < infinity and the solid core for 1 < p < 2 of weighted Bergman spaces A(mu)(p) , 1 < p < infinity, of analytic functions on the disk and on the whole complex plane, for a very general class of nonatomic positive bounded Borel measures mu . New examples are presented. Moreover, we show that the space A(mu)(p) , 1 < p < infinity, is solid if and only if the monomials are an unconditional basis of this space., Bonet's work was partially supported by the MTM2016-76647-P and GV Prometeo/2017/102 projects (Spain). Taskinen's work was partially supported by a research grant from the Faculty of Science of the University of Helsinki.

[EN] We determine the solid hull and solid core of weighted Banach spaces H-upsilon(infinity) of analytic functions functions f such that upsilon vertical bar f vertical bar is bounded, both in the case of the holomorphic functions on the disc and on the whole complex plane, for a very general class of radial weights upsilon. Precise results are presented for concrete weights on the disc that could not be treated before. It is also shown that if H-upsilon(infinity) is solid, then the monomials are an (unconditional) basis of the closure of the polynomials in H-upsilon(infinity). As a consequence H-upsilon(infinity) does not coincide with its solid hull and core in the case of the disc. An example shows that this does not hold for weighted spaces of entire functions., The research of Bonet was partially supported by the project MTM2016-76647-P. The research of Taskinen was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

We consider moderately growing weight functions $v$ on the upper half plane $\mathbb G$ called normal weights which include the examples $(\mathrm{Im} w)^a$, $w \in \mathbb G$, for fixed $a > 0$. In contrast to the comparable, well-studied situation of normal weights on the unit disc here there are always unbounded composition operators $C_{\varphi }$ on the weighted spaces $Hv(\mathbb G)$. We characterize those holomorphic functions $\varphi \colon \mathbb G \rightarrow \mathbb G$ where the composition operator $C_{\varphi } $ is a bounded operator $Hv(\mathbb G) \rightarrow Hv(\mathbb G)$ by a simple property which depends only on $\varphi $ but not on $v$. Moreover we show that there are no compact composition operators $C_{\varphi }$ on $Hv(\mathbb G)$.

[EN] Let v be a radial weight function on the unit disc or on the complex plane. It is shown that for each analytic function f0 in the Banach space Hv all analytic functions f such that v|f| is bounded, the distance of f0 to the subspace Hv0 of Hv of all the functions g such that v|g| vanishes at infinity is attained at a function g0Hv0. Moreover a simple, direct proof of the formula of the distance of f to Hv0 due to Perfekt is presented. As a consequence the corresponding results for weighted Bloch spaces are obtained., The authors are very thankful to the referees for their careful reading of the manuscript and their suggestions. The research of Bonet was partially supported by the Projects MTM2016-76647-P and GV Prometeo 2017/102. The research of Taskinen was partially supported by the research grant from the Faculty of Science of the University of Helsinki.

[EN] We prove sufficient conditions for the boundedness and compactness of Toeplitz operators T-a in weighted sup-normed Banach spaces H-v(infinity) of holomorphic functions defined on the open unit disc D of the complex plane; both the weights v and symbols a are assumed to be radial functions on D. In an earlier work by the authors was shown that there exists a bounded, harmonic (thus non-radial) symbol a such that T-a is not bounded in any space H-v(infinity) with an admissible weight v. Here, we show that a mild additional assumption on the logarithmic decay rate of a radial symbol a at the boundary of D guarantees the boundedness of T-a. The sufficient conditions for the boundedness and compactness of T-a, in a number of variations, are derived from the general, abstract necessary and sufficient condition recently found by the authors. The results apply for a large class of weights satisfying the so called condition (B), which includes in addition to standard weight classes also many rapidly decreasing weights. (c) 2020 Elsevier Inc. All rights reserved., The research of Bonet was partially supported by the projects MTM2016-76647-P and GV Prometeo/2017/102

We characterize the boundedness and compactness of Toeplitz operators T a with radial symbols a in weighted H ∞ -spaces H v ∞ on the open unit disc of the complex plane. The weights v are also assumed radial and to satisfy the condition (B) introduced by the second named author. The main technique uses Taylor coefficient multipliers, and the results are first proved for them. We formulate a related sufficient condition for the boundedness and compactness of Toeplitz operators in reflexive weighted Bergman spaces on the disc. We also construct a bounded harmonic symbol f such that T f is not bounded in H v ∞ for any v satisfying mild assumptions. As a corollary, the Bergman projection is never bounded with respect to the corresponding weighted sup-norms. However, we also show that, for normal weights v, all Toeplitz operators with a trigonometric polynomial as the symbol are bounded on H v ∞ .

By an \emph{assignment} we mean a mapping from a Choquet simplex $K$ to probability measure-preserving systems, obeying some natural restrictions. We prove that if $\Phi$ is an aperiodic assignment on a Choquet simplex $K$ such that the set of extreme points $\mathsf{ex}K$ is a countable union $\bigcup_n E_n$, where each set $E_n$ is compact, zero-dimensional, and the restriction of $\Phi$ to the Bauer simplex $K_n$ spanned by $E_n$ can be `embedded' in some topological dynamical system, then $\Phi$ can be `realized' in a zero-dimensional system., Comment: 15 pages, 1 figure

[EN] We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space R-N. We show that given a weighted L-p-space L-w(p)(R-N) with 1 0 such that, if the initial data f belong to the closed linear space of e(n) with n >= n(m), then the decay rate of the solution of the heat equation is at least t(-m). Such a basis can be constructed as a perturbation of any given Schauder basis. The proof is based on a construction of a basis of L-w(p)(R-N), which annihilates an infinite sequence of bounded functionals., Open access funding provided by University of Helsinki including Helsinki University Central Hospital. The authors would like to thank Thierry Gallay (Grenoble) for discussions which helped in the final formulation of our results. The research of Bonet was partially supported by the projects MTM2016-76647-P and GV Prometeo 2017/102. The research of Taskinen was partially supported by the research grant from the Faculty of Science of the University of Helsinki.

[EN] It is shown that the monomials A = (z(n))(n=0)(infinity) are a Schauder basis of the Frechet spaces A(+)(-gamma), gamma >= 0, that consists of all the analytic functions f on the unit disc such that (1 - vertical bar z vertical bar)(mu)vertical bar f(z)vertical bar is bounded for all mu > gamma. Lusky proved that A is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type H-infinity. A sequence space representation of the Frechet space A(+)(-gamma) is presented. The case of (LB)-spaces A(-)(-gamma), gamma > 0, that are defined as unions of weighted Banach spaces is also studied., Research of the third author was partially supported by the Vaisala Foundation of the Finnish Academy of Sciences and Letters.

Let v be a standard weight on the upper half-plane G, i.e. v : G → ]0,∞[ is continuous and satisfies v(w) = v(i Im w), w ∈ G, v(it) ≥ v(is) if t ≥ s > 0 and limt→0 v(it) = 0. Put v1(w) = Im w v(w), w ∈ G. Then we characterize boundedness and surjectivity of the differentiation operator D : Hv(G) → Hv1(G). For example we show that D is bounded if and only if v is at most of moderate growth. We also study composition operators on Hv(G).

We study Toeplitz operators Ta with radial symbols in weighted Bergman spaces Aμ, 1 < p < ∞, on the disc. Using a decomposition of Aμ into finite dimensional subspaces the operator Ta can be considered as a coefficient multiplier. This leads to new results on boundedness of Ta and also shows a connection to Hardy space multipliers. Using another method we also prove a necessary and suffient condition for the boundedness of Ta for a satisfying an assumption on the positivity of certain indefinite integrals.

Let Un be the unit polydisk in Cn and S be the space of functions of regular variation. Let 1 ≤ p < ∞, ω = (ω1, ..., ωn), ωj ∈ S(1 ≤ j ≤ n) and f ∈ H(Un). The function f is said to be in holomorphic Besov space Bp(ω) if $$ \left\| f \right\|_{B_p (\omega )}^p = \int_{U^n } {\left| {Df(z)} \right|^p \prod\limits_{j = 1}^n {\frac{{\omega _j (1 - |z_j |)}} {{(1 - |z_j |^{2 - p} )}}} dm_{2n} (z) < + \infty } $$ where dm2n(z) is the 2n-dimensional Lebesgue measure on Un and D stands for the fractional differentation of f. This work gives a complete description of (Bp(ω))*, where X* means the dual space of X.. Also the problem of diagonal mapping is completely solved.

We generalize the results of [11] and [12] for the unit ball \( \mathbb{B}_d \) of ℂd. In particular, we show that under the weight condition (B) the weighted H∞-space on \( \mathbb{B}_d \) is isomorphic to l∞ and thus complemented in the corresponding weighted L∞-space. We construct concrete, generalized Bergman projections accordingly. We also consider the case where the domain is the entire space ℂd. In addition, we show that for the polydisc \( \mathbb{D}^d \)d, the weighted H∞-space is never isomorphic to l∞.

Assuming that S is the space of functions of regular variation and ω = (ω1, …, ω n ), ω j ∈ S, by A p (ω) we denote the class of all holomorphic functions defined on the polydisc U n such that where dm 2n (z) is the 2n-dimensional Lebesgue measure on U n . In this article, we characterize the multipliers of A p (ω) (0 < p ≤ 1) and study the boundedness of Toeplitz operators on this space. As an application, a theorem about the division by good-inner functions in A p (ω) is proved. We also consider the generalized little Hankel and Berezin-type operators on A p (ω) (and on L 1(ω)) and prove some theorems about the boundedness of these operators.

We give necessary and sufficient conditions on the weightsv and w such that the differentiation operator D : Hv() ! Hw() between two weighted spaces of holomorphic functions is bounded and onto. Here = C or = D. In particular we characterize all weights v such that D : Hv() ! Hw() is bounded and onto where w(r) = v(r)(1 r) if = D and w = v if = C. This leads to a new description of normal weights.

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of Hv = { f : Ω → C holomorphic : sup z∈Ω |f(z)|v(z) < ∞ } and investigate some isomorphism classes of hv = { f : Ω → C harmonic : sup z∈Ω |f(z)|v(z) < ∞ } where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either Hv ∼ l∞ or Hv ∼ H∞, and at least two possibilities for hv, again hv ∼ l∞ and hv ∼ H∞. We also discuss many new examples of weights.

LetX be a Banach space with a sequence of linear, bounded finite rank operatorsR n:X→X such thatR nRm=Rmin(n,m) ifn≠m and lim n→∞ R n x=x for allx∈X. We prove that, ifR n−Rn −1 factors uniformly through somel p and satisfies a certain additional symmetry condition, thenX has an unconditional basis. As an application, we study conditions on Λ ⊂ ℤ such thatL Λ=closed span $$\left\{ {z^k :k \in \Lambda } \right\} \subset L_1 \left( \mathbb{T} \right)$$ , where $$\mathbb{T} = \left\{ {z \in \mathbb{C}:\left| z \right| = 1} \right\}$$ , has an unconditional basis. Examples include the Hardy space $$H_1 = L_{\mathbb{Z}_ + } $$ .

Let fRng 1=1 be a commuting approximating sequence of the Banach space X leaving the closed subspace A X invariant. Then we prove three-space results of the following kind: If the operators Rn induce basis projections on X=A, and X or A is anLp-space, then both X and A have bases. We apply these results to show that the spaces C = spanfz k : k2 g C(T) and L = spanfz k : k2 g L1(T) have bases whenever Z and Zn is a Sidon set.

The Fourier series of the elements in the generalized Bergman spaces b p , q of harmonic functions over D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = { r [mid ] k [mid ] e ik ϕ } k ∈ℤ is never a basis of b 1, 1 and b ∞, 0 for any weighted L 1 -norm and L ∞ -norm over D . The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted L 1 -norms and L ∞ -norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L 1 -norms and L ∞ -norms over [Copf ] where Ω is indeed a basis of b 1, 1 and b ∞, 0 . Moreover, using similar methods, a weight is constructed on D where b ∞, ∞ is not isomorphic to l ∞ which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.

We discuss and prove three space properties and basis extension theorems of the following kind: LetY be a separableL 1-space andX⊄Y a non-reflexive subspace such thatY/X has a basis. ThenX has a basis. IfY is a separableC(K)-space andX⊄Y is such thatY/X is nonreflexive, then every basis ofX can be extended to a basis ofY.

We show that a Banach space X has a basis provided there are bounded linear finite rank operators R n : X → X such that lim n R n x = x for all x ∈ X , R m R n = R min( m , n ) if m ≠ n , and R n − R n −1 factors uniformly through l m n p 's for some p . As an application we obtain conditions on a subset Λ ⊂ Z such that C Λ =closed span{ z k : k ∈ Λ }⊂ C ( T ) and L Λ =closed span{ z k : k ∈ Λ }⊂ L 1 ( T ) have bases.

1 Let D be the open unit disc and μ a positive bounded measure on [0, 1]. Extending results of Mateljevic/Pavlovic and Shields/Williams we give Banach-space descriptions of the classes of all harmonic (holomorphic) functions f : D → C satisfying

Weighted spaces of harmonic and holomorphic functions on the unit disc are studied. We show that for all radial weights which are not decreasing too fast the space of harmonic functions is isomorphic to c0. For the weights that we consider we completely characterize those spaces of holomorphic functions which are isomorphic to c0. Moreover, we determine when the Riesz projection, mapping the weighted space of harmonic functions onto the corresponding space of holomorphic functions, is bounded.

We consider spaces of $2\pi$-periodic holomorphic functions $f$ on the upper halfplane $G$ which are bounded by a~weighted sup-norm $\sup_{w \in G} |f(w)|v(w)$. Here $v: G \rightarrow ]0, \infty[$ is a function which depends essentially only on $Im(w)$, $w \in G$, and satisfies $ \lim_{t \rightarrow 0} v(it) =0$. We give a complete isomorphic classification of such spaces and investigate composition operators and the differentiation operator between them.

Weighted spaces of harmonic and holomorphic functions on the unit disc are discussed. It is shown that these spaces are always subspaces of c0. Moreover, for many weights, it is shown that the weighted space of holomorphic functions has a basis.

Problems of isomorphisms and isometries play a central role in the theory of normed spaces. While two isomorphic normed spaces are identical from the linear point of view two isometric normed spaces are identical from a geometric point if view.

The results of this section were obtained by the first-named author and B.Shekhtman. They appear here for the first time.

Here we study, among other things, completeness and minimality of sequences in Banach spaces. This includes a discussion of bases and finite dimensional Schauder decompositions. Moreover this gives rise to the introduction of uniformly convex and uniformly smooth Banach spaces.

In this chapter we continue the analysis of Muntz polynomials f(t) = Σk=1nαktλk where 0 < λ1 < λ2

In this chapter we study more general classes of subspaces of C[0, 1] which contain the class of Muntz spaces and we discuss theorems on compactness (11.1.2 and 11.2.6) as well as on interpolation (11.3.1). Moreover, we investigate which of these spaces can be embedded into c0. Finally we deal with notions of universality for Muntz sequences and spaces.

In this chapter we discuss how two or more subspaces in a Banach space affect each other by their position in a Banach space and we give applications in the geometry of Banach spaces.

A Banach space U is called universal (for all separable Banach spaces) if for each separable Banach space X there is a subspace Y in U such that X is isometric to Y .

Throughout this section let X be a separable Banach space. We investigate certain approximation properties for X and deal with the question under which additional condition then X has a basis. In this context we also give a sufficient criterion for X to be a dual Banach space.

We consider a fairly large class of Hilbert spaces of holomorphic functions in one complex variable which contains the Hardy, the Fock and the Bergman space. The purpose of our paper is to study linear bounded operators on these spaces by means of their matrix elements (Tek,ej) where ek are the normalized monomials. To this end we introduce a “Fourier series expansion” of T into special operators and discuss how the properties of T are reflected in its “Fourier summands”. We characterize, among others, the essential spectrum and the essential norm for a large class of T including many Toeplitz operators.

We discuss weighted spaces $Hv(\mathbf{G})$ of holomorphic functions on the upper halfplane $\mathbf{G}$ where $v(w) = v(i \operatorname{Im} w)$, $w \in \mathbf{G}$, $\lim_{t\to 0} v(it)=0$ and $v(it)$ is increasing in $t$. We characterize those weights $v$ with moderate growth where $Hv(\mathbf{G})$ is isomorphic to $l_{\infty}$ and we show that this is never the case if $v$ is bounded.