Volterra Type Operators Research Articles

On the radicality property for spaces of symbols of bounded Volterra operators

In [1] it is shown that the Bloch space B in the unit disc has the following radicality property: if an analytic function g satisfies that gn∈B, then gm∈B, for all m≤n. Since B coincides with the space T(Aαp) of analytic symbols g such that the Volterra-type operator Tgf(z)=∫0zf(ζ)g′(ζ)dζ is bounded on the classical weighted Bergman space Aαp, the radicality property was used to study the composition of paraproducts Tg and Sgf=Tfg on Aαp. Motivated by this fact, we prove that T(Aωp) also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of T(Aωp) for a general radial weight, induces us to prove the radicality property for Aωp from precise norm-operator results for compositions of analytic paraproducts.

Optimal Control Problems for Evolutionary Variational Inequalities with Volterra-Type Operators

Optimal Control Problems for Evolutionary Variational Inequalities with Volterra-Type Operators

A Carleson type measure and a family of Möbius invariant function spaces

For 0<s<1, let {zn} be a sequence in the open unit disk such that ∑n(1−|zn|2)sδzn is an s-Carleson measure. In this paper, we consider the connections between this s-Carleson measure and the theory of Möbius invariant F(p, p-2, s) spaces by the Volterra type operator, the reciprocal of a Blaschke product, and second order complex differential equations having a prescribed zero sequence.

Volterra-type operators mapping weighted Dirichlet space into H^infty

The problem of describing the analytic functions g on the unit disc such that the integral operator T_g(f)(z)=int _0^zf(zeta )g'(zeta ),dzeta is bounded (or compact) from a Banach space (or complete metric space) X of analytic functions to the Hardy space H^infty is a tough problem, and remains unsettled in many cases. For analytic functions g with non-negative Maclaurin coefficients, we describe the boundedness and compactness of T_g acting from a weighted Dirichlet space D^p_omega , induced by an upper doubling weight omega , to H^infty . We also characterize, in terms of neat conditions on omega , the upper doubling weights for which T_g: D^p_omega rightarrow H^infty is bounded (or compact) only if g is constant.

Correction to: Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

Correction to: Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

Volterra-type operators on the minimal Möbius-invariant space

AbstractIn this note, we mainly study operator-theoretic properties on the Besov space $B_{1}$ on the unit disk. This space is the minimal Möbius-invariant space. First, we consider the boundedness of Volterra-type operators. Second, we prove that Volterra-type operators belong to the Deddens algebra of a composition operator. Third, we obtain estimates for the essential norm of Volterra-type operators. Finally, we give a complete characterization of the spectrum of Volterra-type operators.

Volterra-Type Operators on the Subclasses of Univalent Functions

Volterra-Type Operators on the Subclasses of Univalent Functions

Volterra-type operator on the subclasses of univalent functions

UDC 517.5 In this article, we examine the necessary and sufficient conditions for a member to belong to the class of starlike and convex functions of complex order and spirallike functions of type with the complex order We obtain sharp estimates for the coefficient of the second term in the Taylor series of functions belonging to the mentioned classes. In the main part of this paper, we obtain the necessary and sufficient conditions of boundedness for the image of the open unit disk under the action of a Volterra-type operator and the product of the composition operator and Volterra-type operator in the space of univalent functions and its subspace. Finally, we obtain an estimate of the Schwartzian norm of the above operators in these spaces.

Semigroups of Composition Operators and Integral Operators in BMOA-Type Spaces

The aim of this article is to study semigroups of composition operators \(T_t=f\circ \phi _t\) on the BMOA-type spaces \(\textit{BMOA}_p\), and on their “little oh” analogues \(\textit{VMOA}_p\). The spaces \(\textit{BMOA}_p\) were introduced by R. Zhao as part of the large family of F(p, q, s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces \(D^p_{p-1}\). We study the maximal subspace \([\phi _t, \textit{BMOA}_p]\) of strong continuity, providing a sufficient condition on the infinitesimal generator of \(\{\phi _t\}\), under which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), and a related necessary condition in the case where the Denjoy–Wolff point of the semigroup is in \({{\mathbb {D}}}\). Further, we characterize those semigroups, for which \([\phi _t, \textit{BMOA}_p]=\textit{VMOA}_p\), in terms of the resolvent operator of the infinitesimal generator of \((T_t|_{\textit{VMOA}_p})\). In addition we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators \(T_g\). We characterize the symbols g for which \(T_g:\,\textit{BMOA}\rightarrow \textit{BMOA}_1\) is bounded or compact, thus extending a related result to the case \(p=1\). We also prove that for \(1<p<2\) compactness of \(T_g\) on \(\textit{BMOA}_p\) is equivalent to weak compactness.

Volterra Type Operators on Weighted Dirichlet Spaces

The Carleson measures for weighted Dirichlet spaces had been characterized by Girela and Peláez, who also characterized the boundedness of Volterra type operators between weighted Dirichlet spaces. However, their characterizations for the boundedness are not complete. In this paper, the author completely characterizes the boundedness and compactness of Volterra type operators from the weighted Dirichlet spaces D pα to D qβ (−1 < α, β and 0 < p < q < ∞), which essentially complete their works. Furthermore, the author investigates the order boundedness of Volterra type operators between weighted Dirichlet spaces.

Strict singularity of Volterra type operators on Hardy spaces

In this paper, we first characterize the boundedness and compactness of Volterra type operator Sgf(z)=∫0zf′(ζ)g(ζ)dζ,z∈D, defined on Hardy spaces Hp,0<p<∞. The spectrum of Sg is also obtained. Then we prove that Sg fixes an isomorphic copy of ℓp and an isomorphic copy of ℓ2 if the operator Sg is not compact on Hp(1≤p<∞). In particular, this implies that the strict singularity of the operator Sg coincides with the compactness of the operator Sg on Hp. At last, we post an open question for further study.

Explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative Volterra type operators

Abstract In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

Products of Volterra Type Operators and Composition Operators Between Fock Spaces

We show that entire functions $$\varphi $$ , which induce bounded products of Volterra integral operators $$V_g$$ (Volterra companion operators $$J_g$$ ) and composition operators $$C_{\varphi }$$ acting between different Fock spaces, must be affine functions, i.e. $$\varphi (z) = az + b$$ . Then, using this special form of $$\varphi $$ , we characterize boundedness and compactness of these products in term of new quantities, which are much simpler than the Berezin type integral transforms in the previous papers.

Вольтерровы функциональные уравнения в проблеме устойчивости существования глобальных решений распределенных управляемых систем

Earlier the author proposed a rather general form of describing controlled initial–boundary value problems (CIBVPs) by means of Volterra functional equations (VFE) z(t)=f(t,A[z](t),v(t) ), t≡{t^1,⋯,t^n }∈Π⊂R^n, z∈L_p^m≡(L_p (Π) )^m, where f(.,.,.):Π×R^l×R^s→R^m; v(.)∈D⊂L_k^s – control function; A:L_p^m (Π)→L_q^l (Π)- linear operator; the operator A is a Volterra operator for some system T of subsets of the set Π in the following sense: for any H∈T, the restriction A├ [z]┤|_H does not depend on the values of ├ z┤|_(Π\H); (this definition of the Volterra operator is a direct multidimensional generalization of the well-known Tikhonov definition of a functional Volterra type operator). Various CIBVP (for nonlinear hyperbolic and parabolic equations, integro-differential equations, equations with delay, etc.) are reduced by the method of conversion the main part to such functional equations. The transition to equivalent VFE-description of CIBVP is adequate to many problems of distributed optimization. In particular, the author proposed (using such description) a scheme for obtaining sufficient stability conditions (under perturbations of control) of the existence of global solutions for CIBVP. The scheme uses continuation local solutions of functional equation (that is, solutions on the sets H∈T). This continuation is realized with the help of the chain {H_1⊂H_2⊂⋯⊂H_(k-1)⊂H_k≡Π}, where H_i∈T, i=¯(1,k.) A special local existence theorem is applied. This theorem is based on the principle of contraction mappings. In the case p=q=k=∞ under natural assumptions, the possibility of applying this principle is provided by the following: the right-hand side operator F_v [z(.) ](t)≡f(t,A[z](t),v(t)) satisfies the Lipschitz condition in the operator form with the quasi-nilpotent «Lipschitz operator». This allows (using well-known results of functional analysis) to introduce in the space L_∞^m (H) such an equivalent norm in which the operator of the right-hand side will be contractive. In the general case 1≤p,q,k ≤∞, (this case covers a much wider class of CIBVP), the operator F_v; as a rule, does not satisfy such Lipschitz condition. From the results obtained by the author earlier, it follows that in this case there also exists an equivalent norm of the space L_p^m (H), for which the operator F_v is a contraction operator. The corresponding basic theorem (equivalent norm theorem) is based on the notion of equipotential quasi-nilpotency of a family of linear operators, acting in a Banach space. This article shows how this theorem can be applied to obtain sufficient stability conditions (under perturbations of control) of the existence of global solutions of VFE.

Volterra-type operators from F(p,q,s) space to Bloch-Orlicz and Zygmund-Orlicz spaces

In this paper, we discussed the equivalent conditions for the boundedness and compactness of several Volterra-type operators acting from general F(p,q,s) space to Bloch-Orlicz and Zygmund-Orlicz spaces.

On the fundamental solution and its application in a large class of differential systems determined by Volterra type operators with delay

The variation-of-constants formula is one of the principal tools of the theory of differential equations. There are many papers dealing with different versions of the variation-of-constants formula and its applications. Our main purpose in this paper is to give a variation-of-constants formula for inhomogeneous linear functional differential systems determined by general Volterra type operators with delay. Our treatment of the delay in the considered systems is completely different from the usual methods. We deal with the representation of the studied Volterra type operators. Some existence and uniqueness theorems are obtained for the studied linear functional differential and integral systems. Finally, some applications are given.

The essential norm of the integral type operators

On the basis of the characterizations of the boundedness and compactness of the Volterra type operator $$I_{g, \varphi }$$ from mixed-norm spaces $$H(p,\, q,\, \phi )$$ to Zygmund spaces $$ \mathcal {Z}$$, the authors provide a function-theoretic estimate for the essential norm of Volterra type operator $$I_{g, \varphi }$$ by means of the definition of the essential norm of an operator and the properties of the analytic function. An estimate for the essential norm of the generalized integration operator $$\begin{aligned} I^{(n)}_{g, \varphi }f(z)=\int ^{z}_{0}f^{(n)}(\varphi (\xi ))g(\xi )d\xi , \ z\in \mathbb {D}, \end{aligned}$$from Bloch-type spaces to F(p, q, s) spaces is also obtained.

Evolutionary Variational Inequalities with Volterra Type Operators

In this paper, we consider the initial-value problem for parabolic variational inequalities (subdifferential inclusions) with Volterra type operators. We prove the existence and the uniqueness of the solution. Furthermore, the estimates of the solution are obtained. The results are achieved using the Banach's fixed point theorem (the principle of compression mappings). The motivation for this work comes from the evolutionary variational inequalities arising in the study of frictionless contact problems for linear viscoelastic materials with long-term memory. Also, such kind of problems have their application in constructing different models of the injection molding processes.

Volterra Type Operators Between Bloch Type Spaces and Weighted Banach Spaces

When the weight $\mu$ is more general than normal, the complete characterizations in terms of the symbol $g$ and weights for the conditions of the boundedness and compactness of $T_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ and $S_g: H^{\infty}_\nu\rightarrow H^{\infty}_\mu$ are still unknown. Smith et al. firstly gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, continuing their lines of investigations, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators $T_g$ and $S_g$ between Bloch type spaces $\mathcal{B}^\infty_\nu$ and weighted Banach spaces $H^{\infty}_\nu$ with more general weights, which generalize their works.

Dark Energy and Dark Matter Interaction: Kernels of Volterra Type and Coincidence Problem

We study a new exactly solvable model of coupling of the Dark Energy and Dark Matter, in the framework of which the kernel of non-gravitational interaction is presented by the integral Volterra-type operator well-known in the classical theory of fading memory. Exact solutions of this isotropic homogeneous cosmological model were classified with respect to the sign of the discriminant of the cubic characteristic polynomial associated with the key equation of the model. Energy-density scalars of the Dark Energy and Dark Matter, the Hubble function and acceleration parameter are presented explicitly; the scale factor is found in quadratures. Asymptotic analysis of the exact solutions has shown that the Big Rip, Little Rip, Pseudo Rip regimes can be realized with the specific choice of guiding parameters of the model. We show that the Coincidence problem can be solved if we consider the memory effect associated with the interactions in the Dark Sector of the universe.