For a scalar matrix $\mathbf a=(a_{m, n})_{m, n=1}^{\infty},$ the Dirichlet series kernel $κ_{\mathbf a}$ is the double Dirichlet series $κ_{\mathbf a}(s, u) = \sum_{m, n =1}^{\infty} a_{m, n}m^{-s} n^{-\bar{u}}$ in the variables $s$ and $\bar{u},$ which is regularly convergent on some right half-plane $\mathbb H_ρ.$ The analytic symbols $A_{n, \mathbf a} = \sum_{m=1}^{\infty}a_{m, n}m^{-s},$ $n \geq 1$ play a central role in the study of the reproducing kernel Hilbert space $\mathscr H_{\mathbf a}$ associated with the positive semi-definite kernel $κ_{\mathbf a}.$ In particular, they form a total subset of $\mathscr H_{\mathbf a}$ and provide the formula $\sum_{n=1}^{\infty}\langle f, A_{n, \mathbf a} \rangle n^{-s},$ $s \in \mathbb H_ρ,$ for $f \in \mathscr H_{\mathbf a}.$ We combine the basic theory of Dirichlet series kernels with the Gelfond-Schneider theorem (Hilbert's seventh problem) to show that any quasi-invariant Dirichlet series kernel $κ_{\mathbf a}(s, u)$ factors as $f(s)\bar{f(u)}$ for some Dirichlet series $f$ on $\mathbb H_ρ.$ In particular, there is no quasi-invariant Dirichlet series kernel $κ_{\mathbf a}$ if the dimension of $\mathscr H_{\mathbf a}$ is bigger than one. This is in strict contrast with the case of the unit disc, where non-factorable quasi-invariant kernels exist in abundance. We further discuss the Dirichlet series kernels $κ_{\mathbf a}$ invariant under the group $\mathscr T$ of translation automorphisms of $\mathbb H_ρ$ and construct a family of densely defined $\mathscr T$-homogeneous operators in $\mathscr H_{\mathbf a},$ whose adjoints are defined only at the zero vector.
We investigate the compactness of composition operators on the Hardy space of Dirichlet series induced by a map $φ(s)=c_0s+φ_0(s)$, where $φ_0$ is a Dirichlet polynomial. Our results depend heavily on the characteristic $c_0$ of $φ$ and, when $c_0=0$, on both the degree of $φ_0$ and its local behaviour near a boundary point. We also study the approximation numbers for some of these operators. Our methods involve geometric estimates of Carleson measures and tools from differential geometry.
For a Dirichlet series g, we study the Volterra operator Tg of symbol g, acting on a class of weighted Hilbert spaces of Dirichlet series. We obtain sufficient / necessary conditions for Tg to be bounded (resp. compact), involving BMO and Bloch type spaces on some half-plane. We also investigate the membership of Tg in Schatten classes. We also relate the boundedness of Tg to the boundedness of a multiplicative Hankel form of symbol g.
We consider composition operators $\mathscr{C}_φ$ on the Hardy space of Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols $φ$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $φ$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $φ$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particula…
In this paper, we introduce a general class of weighted spaces of holomorphic Dirichlet series (with real frequencies) analytic in some half-plane and study composition operators on these spaces. In the particular case when the symbol inducing the composition operator is an affine function, we give criteria for boundedness and compactness. We also study the cyclicity property and as a byproduct give a sufficient condition so that the direct sum of the identity plus a weighted forward shift operator on the Hardy space H^2 is cyclic.
Let $Φ$ and $Ψ$ be frames for $\cal H$ and let $M_{m,Φ,Ψ}$ be a frame multiplier with the symbol $m$. In this paper, we restrict our investigation to show that the operator properties of $M_{m,Φ,Ψ}$ are stable under the perturbations of $Φ$, $Ψ$ and $m$. Also, special attention is devoted to the study of invertible frame multipliers. These results are not only of interest in their own right, but also they pave the way for obtaining some new results for Gabor multipliers which have been studied mostly by Hans Georg Feichtinger and his coauthors in recent years.
We show that the symbol of a bounded composition operator on the Wiener algebra of Dirichlet series does not need to belong to this algebra. Our example even gives an absolutely summing (hence compact) composition operator.
For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, the associated Volterra operator $\mathbf{T}_g$ acting on a Dirichlet series $f(s)=\sum_{n\ge 1} a_n n^{-s}$ is defined by the integral $f\mapsto -\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p < \infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\operatorname{BMOA…
In this paper, we study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. We study the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $\mathscr{C}_φ$ on $\mathscr{H}^2$ which are generated by symbols of the form $φ(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $φ$, then the operator $\mathscr{C}_φ$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $\mathscr{C}_φ$. Furthermore, in the…
The aim of this paper is to study when two composition operators on the Hilbert space of Dirichlet series with square summable coefficients belong to the same component or when their difference is compact. As a corollary we show that if a linear combination of composition operators with polynomial symbols of degree at most 2 is compact, then each composition operator is compact.
We study the composition operators on an algebra of Dirichlet series, the analogue of the Wiener algebra of absolutely convergent Taylor series, which we call the Wiener-Dirichlet algebra. The central issue is to understand the connection between the properties of the operator and of its symbol, with special emphasis on the compact, automorphic, or isometric character of this operator. We are led to the intermediate study of algebras of functions of several, or countably many, complex variables.
By using the Schur test, we give some upper and lower estimates on the norm of a composition operator on $\mathcal{H}^2$, the space of Dirichlet series with square summable coefficients, for the inducing symbol $φ(s)=c_1+c_{q}q^{-s}$ where $q\geq 2$ is a fixed integer. We also give an estimate on the approximation numbers of such an operator.
We study the decay of approximation numbers of compact composition operators on the Dirichlet space. We give upper and lower bounds for these numbers. In particular, we improve on a result of O. El-Fallah, K. Kellay, M. Shabankhah and A. Youssfi, on the set of contact points with the unit circle of a compact symbolic composition operator acting on the Dirichlet space D. We extend their results in two directions: first, the contact only takes place at the point 1. Moreover, the approximation numbers of the operator can be arbitrarily sub-exponentially small.
For a partition $\boldsymbol{k} = (k_1, \dots, k_m)$ of $n$ consider the group $\mathrm{U}(\boldsymbol{k}) = \mathrm{U}(k_1) \times \dots \times \mathrm{U}(k_m)$ block diagonally embedded in $\mathrm{U}(n)$ and the center $\mathbb{T}^m$ of $\mathrm{U}(\boldsymbol{k})$. We study the Toeplitz operators with $\mathbb{T}^m$-invariant symbols acting on the weighted Bergman spaces on the unit ball $\mathbb{B}^n$. We introduce the $(\boldsymbol{k},j)$-quasi-radial quasi-homogeneous symbols as those that are invariant under the group $\mathrm{U}(\boldsymbol{k},j,\mathbb{T})$ obtained from $\mathrm{U}(…
Some Properties of Products of Linear Fractional Composition Operators on Weighted Dirichlet Spaces
School of Mathematics and Information Science, Guangzhou University（广州大学数学与信息科学学院）；College of Mathematics and Statistics, Chongqing University（重庆大学数学与统计学院）
We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy--Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described b…
We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurli…
Let $Ω$ be a bounded domain in R d with Lipschitz boundary $Γ$. We define the Dirichlet-to-Neumann operator N on L 2 ($Γ$) associated with a second order elliptic operator A = -- d k,j=1 $\partial$ k (c kl $\partial$ l) + d k=1 b k $\partial$ k -- $\partial$ k (c k $\times$) + a 0. We prove a criterion for invariance of a closed convex set under the action of the semigroup of N. Roughly speaking, it says that if the semigroup generated by --A, endowed with Neumann boundary conditions, leaves invariant a closed convex set of L 2 ($Ω$), then the 'trace' of this convex set is invariant for the se…
We observe that local embedding problems for certain Hardy and Bergman spaces of Dirichlet series are equivalent to boundedness of a class of composition operators. Following this, we perform a careful study of such composition operators generated by polynomial symbols $φ$ on a scale of Bergman--type Hilbert spaces $\mathcal{D}_α$. We investigate the optimal $β$ such that the composition operator $\mathcal{C}_φ$ maps $\mathcal{D}_α$ boundedly into $\mathcal{D}_β$. We also prove a new embedding theorem for the non-Hilbertian Hardy space $\mathcal H^p$ into a Bergman space in the half-plane and …
We consider a class of pseudodifferential operators defined on the product of two closed manifolds, with crossed vector valued symbols. We study the asymptotic expansion of Weyl counting function of positive selfadjoint operators in this class. Exploiting a general Theorem of J. Aramaki, we determine the first term of the asymptotic expansion of Weyl counting function and, in a special case, we find the second term. We finish with some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet Divisor Problem.
The paper treats boundary value problems for the fractional Laplacian $(-Δ)^a$, $a>0$, and more generally for classical pseudodifferential operators ($ψ$do's) $P$ of order $2a$ with even symbol, applied to functions on a smooth subset $Ω$ of ${\Bbb R}^n$. There are several meaningful local boundary conditions, such as the Dirichlet and Neumann conditions $γ_k^{a-1}u=φ$, $k=0,1$, where $γ_k^{a-1}u=c_k\partial_n^k(u/d^{a-1})|_{\partialΩ}$, $d(x)=\operatorname{dist}(x,\partialΩ)$. We show a new Green's formula $$(Pu,v)_Ω-(u,P^*v)_Ω=(s_0γ_1^{a-1}u+Bγ_0^{a-1}u,γ_0^{a-1}v)_{\partialΩ}-(s_0γ_0^{a-1}u…
Boundedness of weighted composition operators $W_{u,φ}$ acting on the classical Dirichlet space $\mathcal{D}$ as $W_{u,φ}f= u\, (f\circ φ)$ is studied in terms of the multiplier space associated to the symbol $φ$, i.e., ${\mathcal{M}(ϕ)}=\{ u \in {\mathcal D}: W_{u,ϕ} \hbox{ is bounded on } {\mathcal D} \}$. A prominent role is played by the multipliers of the Dirichlet space. As a consequence, the spectrum of $W_{u,φ}$ in $\mathcal{D}$ whenever $φ$ is an automorphism of the unit disc is studied, extending a recent work of Hyvärinen, Lindström, Nieminen and Saukko to the context of the Dirichl…
For $0<p<\infty $, the Dirichlet-type space $\Dp$ consists of those analytic functions $f$ in the unit disc $\D$ such that $\int_\D|f'(z)|\sp p(1-|z|)^{p-1}\,dA(z)<\infty$. Motivated by operator theoretic differences between the Hardy space $H^p$ and $\Dp$, the integral operator {displaymath} T_g(f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ,\quad z\in\D, {displaymath} acting from one of these spaces to another is studied. In particular, it is shown, on one hand, that $T_g:\Dp\to H^p$ is bounded if and only if $g\in\BMOA$ when $0<p\le 2$, and, on the other hand, that this equivalence is very far from bein…
This paper is devoted to the study of reducing subspaces for multiplication operator $M_ϕ$ on the Dirichlet space with symbol of finite Blaschke product. The reducing subspaces of $M_ϕ$ on the Dirichlet space and Bergman space are related. Our strategy is to use local inverses and Riemann surface to study the reducing subspaces of $M_ϕ$ on the Bergman space, and we discover a new way to study the Riemann surface for $ϕ^{-1}\circϕ$. By this means, we determine the reducing subspaces of $M_ϕ$ on the Dirichlet space when the order of $ϕ$ is $5$; $6$; $7$ and answer some questions of Douglas-Putin…
For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.
We construct a Weyl pseudodifferential calculus tailored to studying boundedness of operators on weighted $L^p$ spaces over $\mathbb{R}^d$ with weights of the form $\exp(-ϕ(x))$, for $ϕ$ a $C^2$ function, a setting in which the operator associated to the weighted Dirichlet form typically has only holomorphic functional calculus. A symbol class giving rise to bounded operators on $L^p$ is determined, and its properties analysed. This theory is used to calculate an upper bounded on the $H^\infty$ angle of relevant operators, and deduces known optimal results in some cases. Finally, the symbol cl…
The distribution $μ_{cl}$ of a Poisson cluster process in $X=\mathbb{R}^{d}$ (with i.i.d. clusters) is studied via an auxiliary Poisson measure on the space of configurations in $\mathfrak{X}=\sqcup_{n} X^n$, with intensity measure defined as a convolution of the background intensity of cluster centres and the probability distribution of a generic cluster. We show that the measure $μ_{cl}$ is quasi-invariant with respect to the group of compactly supported diffeomorphisms of $X$ and prove an integration-by-parts formula for $μ_{cl}$. The corresponding equilibrium stochastic dynamics is then co…
We study composition operators on Hardy and Dirichlet spaces belonging to Schatten classes. We give some new examples and analyse the size of contact set of the symbol of such operators.
In this paper we discuss some spectral invariance results for non-smooth pseudodifferential operators with coefficients in Hölder spaces. In analogy to the proof in the smooth case of Beals and Ueberberg, we use the characterization of non-smooth pseudodifferential operators to get such a result. The main new difficulties are the limited mapping properties of pseudodifferential operators with non-smooth symbols and the fact, that in general the composition of two non-smooth pseudodifferential operators is not a pseudodifferential operator. In order to improve these spectral invariance results …
Using the C* algebraic scattering approach to study quasifree fermionic systems out of equilibrium in quantum statistical mechanics, we construct the nonequilibrium steady state in the isotropic XY chain whose translation invariance has been broken by a local magnetization and analyze the asymptotic behavior of the expectation value for a class of spatial correlation observables in this state. The effect of the breaking of translation invariance is twofold. Mathematically, the finite rank perturbation not only regularizes the scalar symbol of the invertible Toeplitz operator generating the lea…
