Angela A. Albanese, Claudio Mele
We determine multiplication and convolution topological algebras for classes of $ω$-ultradifferentiable functions of Beurling type. Hypocontinuity and discontinuity of the multiplication and convolution mappings are also investigated.
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A characterization of the scalar type spectral generators of Beurling type Carleman ultradifferentiable $C_0$-semigroups is established, the important case of the Gevrey ultradifferentiability is considered in detail, the implementation of the general criterion corresponding to a certain rapidly growing defining sequence is observed.
We prove in a uniform way that all ultradifferentiable function classes of Roumieu- and of Beurling-type defined in terms of a weight matrix admit a convenient setting if the matrix satisfies some mild regularity conditions. We prove that these categories are cartesian closed and as special cases one obtains the classes defined by a weight sequence and by a weight function.
We use techniques from time-frequency analysis to show that the space $\mathcal S_ω$ of rapidly decreasing $ω$-ultradifferentiable functions is nuclear for every weight function $ω(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $\mathcal S_{(M_p)}$ when defined with $L^{2}\,$norms is nuclear exactly when condition $(M2)'$ of Komatsu holds.
In this note we consider weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work [4]. We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main Theorem 5.1 of [4]. In particular, we obtain that the multiplier space of the Gelfand-Shilov space $Σ^{r}_{s}(\mathbb{R}^{d})$ of Beurling type is ultrabornological, whereas the one of the Gelfand-Shilov space $\mathcal{S}^{r}_{s}(\mathbb{R}^{d})$ of Roumieu type …
Quasinormability and property $(Ω)$ for spaces of smooth and ultradifferentiable vectors associated with Lie group representations
本文证明若Fréchet空间E是拟范可积的，则与李群表示相关的光滑和超可微向量空间也是拟范可积的，类似结论对拓扑不变量(Ω)成立，并应用于李群上由权函数族定义的光滑和超可微函数空间。
We prove that functions with compact support in non-quasianalytic classes of Roumieu-type and of Beurling-type defined by a weight matrix with some mild regularity conditions can be characterized by the decay properties of their Fourier transform. For this we introduce the abstract technique of constructing from the original matrix multi-index matrices and associated function spaces. We study the behaviour of this construction in detail and characterize its stability. Moreover non-quasianalyticity of the classes is characterized.
For the ultradifferentiable weight sequence setting it is known that the Borel map which assigns to each function the infinite jet of derivatives (at 0) is surjective onto the corresponding weighted sequence class if and only if the sequence is strongly nonquasianalytic for both the Roumieu- and Beurling-type classes. Sequences which are nonquasianalytic but not strongly nonquasianalytic admit a controlled loss of regularity and we determine the maximal sequence for which such a mixed setting is possible for both types, hence get information on the controlled loss of surjectivity in this situa…
We prove a version of Whitney's extension theorem in the ultradifferentiable Beurling setting with controlled loss of regularity. As a by-product we show the existence of continuous linear extension operators on certain spaces of Whitney ultrajets on arbitrary closed sets in $\mathbb R^n$.
We characterize stability under composition of ultradifferentiable classes defined by weight sequences $M$, by weight functions $ω$, and, more generally, by weight matrices $\mathfrak{M}$, and investigate continuity of composition $(g,f) \mapsto f \circ g$. In addition, we represent the Beurling space $\mathcal{E}^{(ω)}$ and the Roumieu space $\mathcal{E}^{\{ω\}}$ as intersection and union of spaces $\mathcal{E}^{(M)}$ and $\mathcal{E}^{\{M\}}$ for associated weight sequences, respectively.
On the inclusion relations of global ultradifferentiable classes defined by weight matrices
本文研究了在一般权重矩阵框架下全局超可微类的包含关系，通过定义权重矩阵的增长关系进行刻画，并构造了在临界权重序列附近振荡的序列，同时获得了权重函数与序列间的比较结果。
Topology and regularity for generalized ultradistribution algebras
该数学研究梳理非拟解析超分布空间与Colombeau型广义超分布代数的核心结果，分析广义超分布的强结合性与正则性的关联，证明弱可忽略网的性质及平移不变广义超分布的常值性，完善广义超分布代数的拓扑与正则性理论。
We obtain a characterization of ${\mathcal S}^{\{M_p\}}_{\{M_p\}}(\mathbb R^n)$ and $\mathcal {S}^{(M_p)}_{(M_p)}(\mathbb {R}^n)$, the general Gelfand-Shilov spaces of ultradifferentiable functions of Roumieu and Beurling type, in terms of decay estimates for the Fourier coefficients of their elements with respect to eigenfunction expansions associated to normal globally elliptic differential operators of Shubin type. Moreover, we show that the eigenfunctions of such operators are absolute Schauder bases for these spaces of ultradifferentiable functions. Our characterization extends earlier re…
Iterates of composition operators on global spaces of ultradifferentiable functions
本文研究多项式复合算子在全局超可微函数类上的迭代行为，刻画了等度连续与幂有界条件，并探讨了预解算子在Gelfand-Shilov空间间的连续性。
The object of this paper is to prove a version of the Beurling-Helson-Lowdenslager invariant subspace theorem for operators on certain Banach spaces of functions on a multiply connected domain in the complex plane. The norms for these spaces are either the usual Lebesgue and Hardy space norms or certain continuous gauge norms. In the Hardy space case the expected corollaries include the characterization of the cyclic vectors as the outer functions in this context, a demonstration that the set of analytic multiplication operators is maximal abelian and reflexive, and a determination of the clos…
On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices
本文引入由权重函数矩阵定义的加权$L^p$型空间，研究其包含关系，并探讨其与常规$L^p$空间的重合性、加权空间的非平凡性及平移不变性，最终应用于特定权重函数矩阵。
The aim of this paper is to introduce and to study the space $\mathcal{O}_{M,ω}(\mathbb{R}^N)$ of the multipliers of the space $\mathcal{S}_ω(\mathbb{R}^N)$ of the $ω$-ultradifferentiable rapidly decreasing functions of Beurling type. We determine various properties of the space $\mathcal{O}_{M,ω}(\mathbb{R}^N)$. Moreover, we define and compare some lc-topologies of which $\mathcal{O}_{M,ω}(\mathbb{R}^N)$ can be naturally endowed.
We study composition operators on global classes of ultradifferentiable functions of Beurling type invariant under Fourier transform. In particular, for the classical Gelfand-Shilov classes $Σ_d,\ d > 1,$ we prove that a necessary condition for the composition operator $f\mapsto f\circ ψ$ to be well defined is the boundedness of $ψ'.$ We find the optimal index $d'$ for which $C_ψ(Σ_d({\mathbb R}))\subset Σ_{d'}({\mathbb R})$ holds for any non-constant polynomial $ψ.$
A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous $\|.\|_1$-dominating normalized gauge norms instead of the classical norms on measurable functions and a Beurling type result has been proved for the operator of multiplication by the coordinate function. In this paper, we generalize the above Beurling type result to the context of multiplication by a finite Blaschke factor $B(z)$ and also derive the common invariant subspaces of $B^2(z)$ and $B^3(z)$. These results lead to a factorization result for all functions in t…
The Borel map takes a smooth function to its infinite jet of derivatives (at zero). We study the restriction of this map to ultradifferentiable classes of Beurling type in a very general setting which encompasses the classical Denjoy-Carleman and Braun-Meise-Taylor classes. More precisely, we characterize when the Borel image of one class covers the sequence space of another class in terms of the two weights that define the classes. We present two independent solutions to this problem, one by reduction to the Roumieu case and the other by dualization of the involved Fréchet spaces, a Phragmén-…
We prove an extension theorem for ultraholomorphic classes defined by so-called Braun-Meise-Taylor weight functions and transfer the proofs from the single weight sequence case from V. Thilliez [28] to the weight function setting. We are following a different approach than the results obtained in [11], more precisely we are working with real methods by applying the ultradifferentiable Whitney-extension theorem. We are treating both the Roumieu and the Beurling case, the latter one is obtained by a reduction from the Roumieu case.
Following Beurling's theorem the natural compressions of the multiplication operator in the classical $L^2$ space are compressions to model spaces and to their orthogonal complements. Two possibly different model spaces are considered hence asymmetric truncated Toeplitz and asymmetric dual truncated Toeplitz operators are investigated. The main purpose of the paper is to characterize operators which intertwine compressions of the unilateral shift.
In a recent paper, we gave a topological description of Colombeau type algebras introducing algebras of sequences with exponential weights. Embeddings of Schwartz' spaces into the Colombeau algebra G are well known, but for ultradistribution and periodic hyperfunction type spaces we give new constructions. We show that the multiplication of regular enough functions (smooth, ultradifferentiable or quasianalytic), embedded into corresponding algebras, is the ordinary multiplication.
We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. In particular, we embed the space of hyperfunctions on the unit circle into a differential algebra in such a way that the multiplication of real analytic functions on the unit circle coincides with their pointwise multiplication. Furthermore, we introduce a notion of regularity in our newly defined algebra…
We prove various Beurling-Lax type theorems, when the classical backward-shift operator is replaced by a general resolvent operator associated with a rational function. We also study connections to the Cuntz relations. An important tool is a new representation result for analytic functions, in terms of composition and multiplication operators associated with a given rational function. Applications to the theory of de Branges-Rovnyak spaces, also in the indefinite metric setting, are given.
In this paper we give global characterisations of Gevrey-Roumieu and Gevrey-Beurling spaces of ultradifferentiable functions on compact Lie groups in terms of the representation theory of the group and the spectrum of the Laplace-Beltrami operator. Furthermore, we characterise their duals, the spaces of corresponding ultradistributions. For the latter, the proof is based on first obtaining the characterisation of their $α$-duals in the sense of Koethe and the theory of sequence spaces. We also give the corresponding characterisations on compact homogeneous spaces.
Consider the multiplication operator $M_{B}$ in $L^2(\T)$, where the symbol $B$ is a finite Blaschke product. In this article, we characterize the commutant of $M_{B}$ in $L^2(\T)$, noting the fact that $L^2(\T)$ is not an RKHS. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\T)$. Moreover, we analyze their properties while keeping the whole Hardy space, model space, and Beurling-type subspaces invariant. Furthermore, we extende…
We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. Also we develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calderón.
We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreo…
We investigate some multiplication properties of Kato-Sobolev spaces by adapting the techniques used in the study of Beurling algebras by Coifman and Meyer. We develop an analytic functional calculus for Kato-Sobolev algebras based on an integral representation formula belonging A. P. Calderón. Also we study the Schatten-von Neumann properties of pseudo-differential operators with symbols in the Kato-Sobolev spaces.
Topologizability and related properties of the iterates of composition operators in Gelfand-Shilov classes
本文研究Gelfand-Shilov类中由多项式定义的复合算子迭代的可拓扑化性质，证明次数大于1的多项式算子可拓扑化，并给出仿射函数情形的完整刻画。
The classical completeness problem raised by Beurling and independently by Wintner asks for which $ψ\in L^2(0,1)$, the dilation system $\{ψ(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $ψ$ is identified with its extension to an odd $2$-periodic function on $\mathbb{R}$. This difficult problem is nowadays commonly called as the Periodic Dilation Completeness Problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over the infinite-dimensional pol…
Considered are operators that leave the set of non-invertible (in the sense of Ehrenpreis) distributions stable. They simultaneously generalise the operation of convolution by a distribution with compact support and the operation of multiplication by a real analytic function; they are here called pseudo-convolutions since they also generalise pseudo-differential operators. (It is shown that the elliptic real analytic pseudo-differential operators leave both the non-invertible and the invertible distributions invariant.) But when the condition of real-analyticity is relaxed, such operators may …
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