Bloch Space Research Articles

Formula omitted]-harmonic Bergman projection on the real hyperbolic ball

We determine precisely when the Bergman projections Pβ are bounded from Lebesgue spaces Lαp to weighted Bergman spaces Bαp of H-harmonic functions on the real hyperbolic ball, and verify a recent conjecture of M. Stoll. We obtain upper estimates for the reproducing kernels of the H-harmonic Bergman spaces Bα2 and their partial derivatives. We also consider the projection from L∞ to the Bloch space B of H-harmonic functions.

Closure of Analytic Functions of the Bounded Mean Oscillation in Logarithmic Bloch Spaces

Closure of Analytic Functions of the Bounded Mean Oscillation in Logarithmic Bloch Spaces

Norm and essential norm of an integral‐type operator from the logarithmic Bloch space to the Bloch‐type space on the unit ball

We calculate norm and essential norm of an integral‐type operator from the logarithmic Bloch space and the little logarithmic Bloch space to Bloch‐type spaces on the unit ball of . We show that there is a one‐parameter class of equivalent norms on the logarithmic Bloch space for which the norms can be calculated.

Inner functions in QK spaces and multipliers

Abstract In this paper, under some conditions on weighted function K, we give some new equivalent characterizations of inner functions in Q K spaces. Meanwhile, we also studied inner functions in Q K spaces as multipliers of C 𝓑(Q K ) ∩ BMOA, where C 𝓑(Q K ) denoted the closure of Q K spaces in Bloch spaces.

Operators of the alfa-Bloch space on the open unit ball of a JB*-triple

"Let $\B_X$ be a bounded symmetric domain realized as the open unit ball of a JB*-triple $X$ which may be infinite dimensional. In this paper, we characterize the bounded weighted composition operators from the Hardy space $H^{\infty}(\mathbb{B}_X)$ into the $\alpha $-Bloch space $\mathcal{B}^\alpha (\B_X)$ on $\mathbb{B}_X$. Later, we show the multiplication operator from $H^{\infty}(\mathbb{B}_X)$ into $\mathcal{B}^\alpha (\B_X)$ is bounded. Also, we give the operator norm of the bounded multiplication operator."

A note on Bloch functions

"We construct a natural linear isomorphism between the little Bloch space B0 and the Banach space c0 of complex null sequences. This paper is written for the special issue of Studia Universitatis Babes-Bolyai Mathematica in memory of Professor Gabriela Kohr."

Variational polaron equations applied to the anisotropic Fröhlich model

Starting from recent advances in the first-principles modeling of polarons, variational polaron equations in the strong-coupling adiabatic approximation are formulated in Bloch space. In this framework, polaron formation energy as well as individual electron, phonon and electron-phonon contributions are obtained. We suggest an efficient gradient-based optimization algorithm and apply these equations to the generalized Fr\"ohlich model with anisotropic non-degenerate electronic bands, both in two- and three-dimensional cases. The effect of the divergence of Fr\"ohlich electron-phonon matrix elements at $\Gamma$-point is treated analytically, improving the convergence with respect to the sampling in reciprocal space. The whole methodology is validated by obtaining the known asymptotic solution of the standard Fr\"ohlich model in isotropic scenario and also by comparing our results with the Gaussian ansatz approach, showing the difference between the numerically exact and Gaussian trial wavefunctions. Additionally, decomposition of the energy into individual terms allows one to recover the Pekar's 1:2:3:4 theorem, which is shown to be valid even in the anisotropic case. We expect that the improvements in the formalism and numerical implementation will be applicable beyond the large polaron hypothesis inherent to Fr\"ohlich model.

Composition operators on Bloch and Hardy type spaces

The main purpose of this paper is to discuss Hardy type spaces, Bloch type spaces and the composition operators of complex-valued harmonic functions. We first establish a sharp estimate of the Lipschitz continuity of complex-valued harmonic functions in Bloch type spaces with respect to the pseudo-hyperbolic metric, which gives an answer to an open problem. Then some classes of composition operators on Bloch and Hardy type spaces will be investigated. The obtained results improve and extend some corresponding known results.

Holomorphic semigroups and Sarason’s characterization of vanishing mean oscillation

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA) is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers, Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (\varphi_{t}) that can replace the semigroup of rotations in Sarason’s theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (\varphi_{t}) . In addition, we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch spaces, and surprisingly enough, we find that the semigroups for which the Bloch version of Sarason’s theorem holds are exactly the same as in the BMOA case.

The geometry of Bloch space in the context of quantum random access codes

We study the communication protocol known as a quantum random access code (QRAC) which encodes n classical bits into m qubits ( $$m<n$$ ) with a probability of recovering any of the initial n bits of at least $$p>\tfrac{1}{2}$$ . Such a code is denoted by (n, m, p)-QRAC. If cooperation is allowed through a shared random string, we call it a QRAC with shared randomness. We prove that for any (n, m, p)-QRAC with shared randomness the parameter p is upper bounded by $$ \tfrac{1}{2}+\tfrac{1}{2}\sqrt{\tfrac{2^{m-1}}{n}}$$ . For $$m=2$$ , this gives a new bound of $$p\le \tfrac{1}{2}+\tfrac{1}{\sqrt{2n}}$$ confirming a conjecture by Imamichi and Raymond (AQIS’18). Our bound implies that the previously known analytical constructions of $$(3,2,\tfrac{1}{2}+\tfrac{1}{\sqrt{6}})$$ - , $$(4,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{2}})$$ - and $$(6,2,\tfrac{1}{2}+\tfrac{1}{2\sqrt{3}})$$ -QRACs are optimal. To obtain our bound, we investigate the geometry of quantum states in the Bloch vector representation and make use of a geometric interpretation of the fact that any two quantum states have a nonnegative overlap.

Ces\xe0ro-like operators acting on spaces of analytic functions

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.

BMO and Hankel Operators on Bergman Space of the Siegel Upper Half-Space

On the setting of the Siegel upper half-space we study the spaces of bounded and vanishing mean oscillations which are defined in terms of the Berezin transform, we use them to characterize bounded and compact Hankel operators on Bergman space. When the symbols are specialized to be holomorphic, we study its corresponding Hankel operators in terms of Bloch space of the Siegel upper half-space.

A new characterization of the closure of Dirichlet type spaces Ds in Bloch spaces and interpolating Blaschke product

In this paper, motivated by Qian, et al [20, 22], we give a new characterization for the closure of the space Ds in the Bloch space.

A Derivative-Hilbert operator acting from Besov spaces into Bloch space

A Derivative-Hilbert operator acting from Besov spaces into Bloch space

Essential norm of composition operators on Harmonic Bloch spaces

In this paper we characterize essential norm of composition operators on the spaces of Harmonic Bloch functions. We obtain some upper and lower bounds for the essential norm of composition operators between Harmonic Bloch function spaces. In some cases we find exact value of the essential norm such as in Theorems 2.7 and 2.11. These results extends the similar results that were proven for composition operators on Bloch spaces.

Mean ergodicity of multiplication operators on the Bloch and Besov spaces

Mean ergodicity of multiplication operators on the Bloch and Besov spaces

A class of integral operators induced by harmonic Bergman-Besov kernels on Lebesgue classes

We provide a full characterization in terms of the six parameters involved the boundedness of all standard weighted integral operators induced by harmonic Bergman-Besov kernels acting between different Lebesgue classes with standard weights on the unit ball of Rn. These operators in some sense generalize the harmonic Bergman-Besov projections. To obtain the necessity conditions, we use a technique that heavily depends on the precise inclusion relations between harmonic Bergman-Besov and weighted Bloch spaces on the unit ball. This fruitful technique is new. It has been used first with holomorphic Bergman-Besov kernels by Kaptano?lu and ?reyen. Methods of the sufficiency proofs we employ are Schur tests or H?lder or Minkowski type inequalities which also make use of estimates of Forelli-Rudin type integrals.

Non‐Diskcyclicity of Bounded Composition Operators on the Little Bloch Space and the Besov Space

In this paper, we show that there are no diskcyclic composition operators on the little Bloch space ℬ0 and the Besov spaces Bp.

Multiplication Operators on the Bloch Space of Infinite Dimensional Bounded Symmetric Domains

Multiplication Operators on the Bloch Space of Infinite Dimensional Bounded Symmetric Domains

Norms of inclusions between some spaces of analytic functions

The inclusions between the Besov spaces B^q, the Bloch space mathcal {B} and the standard weighted Bergman spaces A^p_{alpha} are completely understood, but the norms of the corresponding inclusion operators are in general unknown. In this work, we compute or estimate asymptotically the norms of these inclusions.