Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

We introduce discrete Cesaro operator on weighted Banach spaces on homogenous trees. We characterize continuity and compactness of these type of operators acting between weighted Banach spaces on homogenous trees. In particular, we prove that discrete Cesaro operator acting on the Banach space of bounded functions on homogenous trees is always continuous, but it is never compact. However, such an operator acting on weighted Banach spaces of bounded functions on homogenous trees may not be continuous for some weight functions. Moreover, we provide plenty of examples of weight functions such that these type of operators acting on the corresponding weighted Banach spaces of bounded functions on homogenous trees are compact.

We characterize the closure of Dirichlet type spaces Dμ in Bloch type spaces. Moreover, we study the boundedness and compactness of composition operators from Bloch type spaces to closures of Dirichlet type spaces Dμ in Bloch type spaces.

Let X be a finite or infinite dimensional complex Banach space. We characterize the bounded weighted composition operators between the Bloch type space and the Hardy space on the unit ball of X, extending several known results for finite dimensional domains.

For a wide class of radial weights we calculate the essential norm of a weighted composition operator \({uC_\varphi}\) on the weighted Banach spaces of analytic functions in terms of the analytic function \({u \colon \mathbb D \to \mathbb C}\) and the nth power of the analytic selfmap \({\varphi}\) of the open unit disc \({\mathbb D}\) . We also apply our result to calculate the essential norm of composition operators acting on Bloch type spaces with general radial weights.

From a unified vision of vector valued solutions in weighted Banach spaces, this paper establishes the existence and uniqueness for space homogeneous Boltzmann bi-linear systems with conservative collisional forms arising in complex gas dynamical structures. This broader vision is directly applied to dilute multi-component gas mixtures composed of both monatomic and polyatomic gases. Such models can be viewed as extensions of scalar Boltzmann binary elastic flows, as much as monatomic gas mixtures with disparate masses and single polyatomic gases, providing a unified approach for vector valued solutions in weighted Banach spaces. Novel aspects of this work include developing the extension of a general ODE theory in vector valued weighted Banach spaces, precise lower bounds for the collision frequency in terms of the weighted Banach norm, energy identities, angular or compact manifold averaging lemmas which provide coerciveness resulting into global in time stability, a new combinatorics estimate for p-binomial forms producing sharper estimates for the k-moments of bi-linear collisional forms. These techniques enable the Cauchy problem improvement that resolves the model with initial data corresponding to strictly positive and bounded initial vector valued mass and total energy, in addition to only a 2+\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2^+$$\\end{document} moment determined by the hard potential rates discrepancy, a result comparable in generality to the classical Cauchy theory of the scalar homogeneous Boltzmann equation.

Let $${\mathbf {B}}_{{\mathbb {X}}}$$ be the open unit ball of a complex Banach space $${\mathbb {X}}$$ , which may beinfinite dimensional. The weighted composition operator and weighted space defined on $${\mathbf {B}}_{{\mathbb {X}}}$$ are introduced. We obtain the boundedness and compactness of the weightedcomposition operator from the Bloch-type spaces to the weighted spaces, and some properties with the Bloch-type spaces are given. Our main results generalize theprevious works on the Euclidean unit ball $${\mathbb {B}}^n$$ to the case of $${\mathbf {B}}_{{\mathbb {X}}}$$ .

Toeplitz operators on Bloch-type spaces in the unit ball of [formula omitted

It is shown that the monomials Λ = ( z n ) n = 0 ∞ \Lambda =(z^n)_{n=0}^{\infty } are a Schauder basis of the Fréchet spaces A + − γ , γ ≥ 0 , A_+^{-\gamma }, \ \gamma \geq 0, that consists of all the analytic functions f f on the unit disc such that ( 1 − | z | ) μ | f ( z ) | (1-|z|)^{\mu }|f(z)| is bounded for all μ > γ \mu > \gamma . Lusky proved that Λ \Lambda is not a Schauder basis for the closure of the polynomials in weighted Banach spaces of analytic functions of type H ∞ H^{\infty } . A sequence space representation of the Fréchet space A + − γ A_+^{-\gamma } is presented. The case of (LB)-spaces A − − γ , γ > 0 , A_{-}^{-\gamma }, \ \gamma > 0, that are defined as unions of weighted Banach spaces is also studied.

Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem

We study the relation between multiplication operators M g with closed range on L a p and M g with closed range on H ∞ . Let p>0. For 0 < α < 1 , suppose g ∈ B α . Then the following are equivalent. M g : L a p → L a p is bounded below. M g : H ∞ → H ∞ is bounded below. 0 $ ]]> lim inf z → ∂D | g ( z ) | > 0 .

We consider m-th order linear, uniformly elliptic equations Lu=f\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}u=f$$\\end{document} with non-smooth coefficients in Banach–Sobolev spaces WXwm(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{X_w}^m (\\Omega )$$\\end{document} generated by weighted Banach Function Spaces (BFS) Xw(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_w (\\Omega )$$\\end{document} on a bounded domain Ω⊂Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset {\\mathbb R}^{n}$$\\end{document}. Supposing boundedness of the Hardy–Littlewood Maximal operator and the Calderón–Zygmund singular integrals in Xw(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_w (\\Omega )$$\\end{document} we obtain solvability in the small in WXwm(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W_{X_w}^m (\\Omega )$$\\end{document} and establish interior Schauder type a priori estimates. These results will be used in order to obtain Fredholmness of the operator L\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {L}$$\\end{document} in Xw(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_w (\\Omega )$$\\end{document}.

We study the incompleteness of exponential system in C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> from a harmonic measure point of view, where C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α</sub> is the weighted Banach space consists of complex continuous functions f defined on the real axis ℝ with vanishing at infinity, in the uniform norm.

It is proved that in Banach algebras generated by two idempotents and, perhaps, by a certain flip operator the standard identity F4 is fulfilled. The maximal ideal space of such algebras is determined and the corresponding symbol is given. By means of local techniques these results are applied to obtain a symbol calculus for singular integral operators with Carleman shift (changing the orientation) in weighted Banach spaces.

In the year 1944 the well-known French mathematician Jean Favard (1902–1965) introduced a discretely defined operator which is a discrete analogue of the familiar Gauss–Weierstrass singular convolution integral. In the present chapter we sketch the history of this approximation operator during the past 70 years by presenting known results on the operator and its various extensions and variants. The first part after the introduction is dedicated to saturation of the classical Favard operator in weighted Banach spaces. Furthermore, we discuss the asymptotic behaviour of a slight generalization \(F_{n,\sigma _{n}}\) of the Favard operator and its Durrmeyer variant \(\tilde{F }_{n,\sigma _{n}}\). In particular, the local rate of convergence when applied to locally smooth functions is considered. The main result of this part consists of the complete asymptotic expansions for the sequences \(\left (F_{n,\sigma _{n}}f\right )\left (x\right )\) and \(\left (\tilde{F } _{n,\sigma _{n}}f\right )\left (x\right )\) as n tends to infinity. Furthermore, these asymptotic expansions are valid also with respect to simultaneous approximation. A further part is devoted to the recent work of several Polish mathematicians on approximation in weighted function spaces. Finally, we define left quasi-interpolants for the Favard operator and its Durrmeyer variant in the sense of Sablonniere.