Holomorphic Symbols Research Articles

Extended Cesàro Operators Between Hardy and Bergman Spaces on the Complex Ball

We characterize those holomorphic symbols g for which the extended Cesaro operator Vg maps the Hardy space Hp(B) into the weighted Bergman space$$ {A}_{\beta}^q(B) $$, 0 −1, on the unit ball B of ℂd.

Essential Norm Estimates for Little Hankel Operators with Anti Holomorphic Symbols on Weighted Bergman Spaces of the Unit Ball

We give estimates for the essential norm of a little Hankel operator with anti holomorphic symbol on weighted Bergman spaces of the unit ball in terms of the Bloch semi-norm of its symbol function.

Integral, Differential and Multiplication Operators on Generalized Fock Spaces

Volterra companion integral and multiplication operators with holomorphic symbols are studied for a large class of generalized Fock spaces on the complex plane $\CC$. The weights defining these spaces are radial and subject to a mild smoothness condition. In addition, we assumed that the weights decay faster than the classical Gaussian weight. One of our main results show that there exists no nontrivial holomorphic symbols $g$ which induce bounded Volterra companion integral $I_g$ and multiplication operators $M_g$ acting between the weighted spaces. We also describe the bounded and compact Volterra-type integral operators $V_g$ acting between $\mathcal{F}_q^\psi$ and $\mathcal{F}_p^\psi$ when at least one of the exponents $p$ or $q$ is infinite, and extend results of Constantin and Pel\'{a}ez for finite exponent cases. Furthermore, we showed that the differential operator $D$ acts in unbounded fashion on these and the classical Fock spaces.

Volterra operators in the edge-calculus

We study the Volterra property of a class of anisotropic pseudo-differential operators on $$\mathbb {R}\times B$$ for a manifold B with edge Y and time-variable t. This exposition belongs to a program for studying parabolicity in such a situation. In the present consideration we establish non-smoothing elements in a subalgebra with anisotropic operator-valued symbols of Mellin type with holomorphic symbols in the complex Mellin covariable from the cone theory, where the covariable $$\tau $$ of t extends to symbols with respect to $$\tau $$ to the lower complex v half-plane. The resulting space of Volterra operators enlarges an approach of Buchholz (Parabolische Pseudodifferentialoperatoren mit operatorwertigen Symbolen. Ph.D. thesis, Universitat Potsdam, 1996) by necessary elements to a new operator algebra containing Volterra parametrices under an appropriate condition of anisotropic ellipticity. Our approach avoids some difficulty in choosing Volterra quantizations in the edge case by generalizing specific achievements from the isotropic edge-calculus, obtained by Seiler (Pseudodifferential calculus on manifolds with non-compact edges, Ph.D. thesis, University of Potsdam, 1997), see also Gil et al. (in: Demuth et al (eds) Mathematical research, vol 100. Akademic Verlag, Berlin, pp 113–137, 1997; Osaka J Math 37:221–260, 2000).

Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space

Let $$\mathbb{B}$$ be the unit ball of a complex Banach space X. In this paper, we generalize the Bloch-type spaces and the little Bloch-type spaces to the open unit ball $$\mathbb{B}$$ by using the radial derivative. Next, we define an extended Cesaro operator Tφ with holomorphic symbol φ and characterize those φ for which Tφ is bounded between the Bloch-type spaces and the little Bloch-type spaces. We also characterize those φ for which Tφ is compact between the Bloch-type spaces and the little Bloch-type spaces under some additional assumption on the symbol φ. When $$\mathbb{B}$$ is the open unit ball of a finite dimensional complex Banach space X, this additional assumption is automatically satisfied.

Essential Norm Estimates for Little Hankel Operators with Anti Holomorphic Symbols on the weighted Bergman Space of the Unit Disk

We give estimates for the essential norm of little Hankel operators with anti holomorphic symbols on weighted Bergman spaces of the unit disk in terms of the Bloch semi norm of its symbol function.

SPECTRAL PROPERTIES OF VOLTERRA-TYPE INTEGRAL OPERATORS ON FOCK-SOBOLEV SPACES

We study some spectral properties of Volterra-type integral operators $V_g$ and $I_g$ with holomorphic symbol $g$ on the Fock--Sobolev spaces $\mathcal{F}_{\psi_m}^p$. We showed that $V_g$ is bounded on $\mathcal{F}_{\psi_m}^p$ if and only if $g$ is a complex polynomial of degree not exceeding two, while compactness of $V_g$ is described by degree of $g$ being not bigger than one. We also identified all those positive numbers $p$ for which the operator $V_g$ belongs to the Schatten $\mathcal{S}_p$ classes. Finally, we characterize the spectrum of $V_g$ in terms of a closed disk of radius twice the coefficient of the highest degree term in a polynomial expansion of $g$.

Bergman kernels, TYZ expansions and Hankel operators on the Kepler manifold

For a class of O(n+1,R) invariant measures on the Kepler manifold possessing finite moments of all orders, we describe the reproducing kernels of the associated Bergman spaces, discuss the corresponding asymptotic expansions of Tian–Yau–Zelditch, and study the relevant Hankel operators with conjugate holomorphic symbols. Related reproducing kernels on the minimal ball are also discussed. Finally, we observe that the Kepler manifold either does not admit balanced metrics, or such metrics are not unique.

Power bounded composition operators on spaces of meromorphic functions

We study composition operators with holomorphic symbols defined on spaces of meromorphic functions, when endowed with their natural locally convex topology. First, we show that such operators are well-defined, continuous and never compact. Then, we study the dynamics and prove that a composition operator is power bounded or mean ergodic if and only if the symbol is a nilpotent element in the group of automorphisms.

Monge–Ampère measures and Poletsky–Stessin Hardy spaces on bounded hyperconvex domains

In this thesis study, we consider Poletsky-Stessin Hardy (PS-Hardy) spaces that are generated by continuous, plurisubharmonic exhaustion functions on hyperconvex domains. In the first part of this study we examine these spaces on domains in the complex plane that are bounded by an analytic Jordan curve. In this setting we focus on PS-Hardy spaces generated by exhaustion functions that have finite Monge-Ampre mass but are not necessarily maximal outside of a compact set. This choice gives us new Banach spaces strictly contained in classical Hardy spaces. We characterize PS-Hardy spaces through their boundary values and we show factorization results analogous to unit disc case. Using functional analysis techniques we prove that the algebra of holomorphic functions which are continuous on the boundary are dense in PS-Hardy spaces. Moreover, we consider the composition operators with holomorphic symbols acting on PS-Hardy spaces and show that contrary to classical case, not all composition operators are bounded on PS-Hardy spaces. In the second part, we study PS-Hardy spaces on polydisc, complex ellipsoid and on strongly convex domains. On complex ellipsoid case, we prove the existence of radial boundary values and then by applying a classical method given by Stein we show the existence of boundary values along admissible approach regions. As an application of this method , we also obtain that polynomials are dense in PS-Hardy spaces on complex ellipsoids. Lastly, we examine the boundedness of composition operators on PS-Hardy spaces on hyperconvex domains in several variables.

Weighted composition operator from Bloch-type space to H^∞ space on the unit ball

In this paper, we characterize those holomorphic symbols u on the unit ball B and holomorphic self-mappings φ of B for which the weighted composition operator uCφ is bounded or compact from Bloch-type space to H∞ space. Mathematics subject classification (2010): Primary 47B33; Secondary 47B38.

Weighted Banach spaces of harmonic functions

We study Banach spaces of harmonic functions on open sets of \(\mathbb R ^{N}\) or \(\mathbb C ^N\) endowed with weighted supremum norms. We investigate the harmonic associated weight defined naturally as the analogue of the holomorphic associated weight introduced by Bierstedt, Bonet, and Taskinen and we compare them. We study composition operators with holomorphic symbol between weighted Banach spaces of pluriharmonic functions characterizing the continuity, the compactness and the essential norm of composition operators among these spaces in terms of associated weights.

Dynamics of composition operators with holomorphic symbol

We unify and extend several results on the dynamic behaviour of composition operators on the space of holomorphic functions on a simply connected plane domain and endowed with the compact open topology. In particular, we show that a composition operator is weakly supercyclic if and only if the algebra it generates consists entirely, except for the null one, of operators that are topologically mixing, Devaney-chaotic, and have a frequently hypercyclic subspace. We also show that all non-zero scalar multiples of powers of a given hypercyclic composition operator share a dense set of common hypercyclic vectors.

Toeplitz Operators on the Dirichlet Space of 𝔹n

We study the algebraic properties of Toeplitz operators on the Dirichlet space of the unit ball 𝔹n. We characterize pluriharmonic symbol for which the corresponding Toeplitz operator is normal or isometric. We also obtain descriptions of conjugate holomorphic symbols of commuting Toeplitz operators. Finally, the commuting problem of Toeplitz operators whose symbols are of the form is studied.

ESSENTIAL NORM OF EXTENDED CESÀRO OPERATORS FROM ONE BERGMAN SPACE TO ANOTHER

AbstractLet Ap(φ) be the pth Bergman space consisting of all holomorphic functions f on the unit ball B of ℂn for which $\|f\|^p_{p,\varphi }= \int _B |f(z)|^p \varphi (z) \,dA(z)\lt +\infty $, where φ is a given normal weight. Let Tg be the extended Cesàro operator with holomorphic symbol g. The essential norm of Tg as an operator from Ap (φ) to Aq (φ) is denoted by $\|T_g\|_{e, A^p (\varphi )\to A^q (\varphi )} $. In this paper it is proved that, for p≤q, with 1/k=(1/p)−(1/q) , where ℜg(z) is the radial derivative of g; and for p>q, with 1/s=(1/q)−(1/p) .

Holomorphic corner symbols

We consider pseudo-differential operators on a manifold M with corners of second order. Locally near a corner point c the space M is modelled on a cone \({B^{\Delta}:=(\overline{\mathbb R}_{+}\times B)/(\{0\}\times B)}\) for a compact manifold B with smooth edge. We establish some essential elements of the corner pseudo-differential calculus locally near c represented as Mellin operators with holomorphic symbols, with values in the ideal of Mellin and Green operators of the edge algebra over B.

Schatten(-Herz) class extended Cesàro operators on Bergman spaces in the unit ball of ${\mathbf {C}}^n$

For a large class of weights ϕ, we characterize the holomorphic symbols g, for which the extended Cesaro operators Tg acting on the weighted Bergman space A 2 ϕ (B) are in the Schatten (or Schatten-Herz) class in the unit ball of C n .

Bilinear forms on the Dirichlet space

Let $\mathcal{D}$ be the classical Dirichlet space, the Hilbert space of holomorphic functions on the disk. Given a holomorphic symbol function $b$ we define the associated Hankel type bilinear form, initially for polynomials f and g, by $T_{b}(f,g):= _{\mathcal{D}} $, where we are looking at the inner product in the space $\mathcal{D}$. We let the norm of $T_{b}$ denotes its norm as a bilinear map from $\mathcal{D}\times\mathcal{D}$ to the complex numbers. We say a function $b$ is in the space $\mathcal{X}$ if the measure $d\mu_{b}:=| b^{\prime}(z)| ^{2}dA$ is a Carleson measure for $\mathcal{D}$ and norm $\mathcal{X}$ by $$ \Vert b\Vert_{\mathcal{X}}:=| b(0)| +\Vert | b^{\prime}(z)| ^{2}dA\Vert_{CM(\mathcal{D})}^{1/2}. $$ Our main result is $T_{b}$ is bounded if and only if $b\in\mathcal{X}$ and $$ \Vert T_{b}\Vert_{\mathcal{D\times D}}\approx\Vert b\Vert_{\mathcal{X}}. $$

Riemann–Hilbert Approach to a Generalised Sine Kernel and Applications

We investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval [-q,q]. We determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its Fredholm determinant. Further, we apply our results to build the resolvent of truncated Wiener--Hopf operators generated by holomorphic symbols. Finally, the leading asymptotics of the Fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models.

Composition operators induced by smooth self-maps of the real or complex unit balls

In this paper, we study the composition operator C Φ with a smooth but not necessarily holomorphic symbol Φ. A necessary and sufficient condition on Φ for C Φ to be bounded on holomorphic (respectively harmonic) weighted Bergman spaces of the unit ball in C n (respectively R n ) is given. The condition is a real version of Wogen's condition for the holomorphic spaces, and a non-vanishing boundary Jacobian condition for the harmonic spaces. We also show certain jump phenomena on the weights for the target spaces for both the holomorphic and harmonic spaces.