Volterra Operators Research Articles

Optimal domain of generalized Volterra operators

For g in BMOA, we consider the generalized Volterra operator Tg acting on Hardy spaces Hp. This article aims to study the largest space of analytic functions, which is mapped by Tg into the Hardy space Hp. We call this space the optimal domain of Tg and we describe its structural properties. Motivation for this comes from the work of G. Curbera and W. Ricker [7] who studied the optimal domain of the classical Cesáro operator.

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.

Simultaneous space–time Hermite wavelet method for time-fractional nonlinear weakly singular integro-partial differential equations

An innovative simultaneous space–time Hermite wavelet method has been developed to solve weakly singular fractional-order nonlinear integro-partial differential equations in one and two dimensions with a focus whose solutions are intermittent in both space and time. The proposed method is based on multi-dimensional Hermite wavelets and the quasilinearization technique. The simultaneous space–time approach does not fully exploit for time-fractional nonlinear weakly singular integro-partial differential equations. Subsequently, the convergence analysis is challenging when the solution depends on the entire time domain (including past and future time), and the governing equation is combined with Volterra and Fredholm integral operators. Considering these challenges, we use the quasilinearization technique to handle the nonlinearity of the problem and reconstruct it to a linear integro-partial differential equation with second-order accuracy. Then, we apply multi-dimensional Hermite wavelets as attractive candidates on the resulting linearized problems to effectively resolve the initial weak singularity at t=0. In addition, the collocation method is used to determine the tensor-based wavelet coefficients within the decomposition domain. We elaborate on constructing the proposed simultaneous space–time Hermite wavelet method and design comprehensive algorithms for their implementation. Specifically, we emphasize the convergence analysis in the framework of the L2 norm and indicate high accuracy dependent on the regularity of the solution. The stability of the proposed wavelet-based numerical approximation is also discussed in the context of fractional-order nonlinear integro-partial differential equations involving both Volterra and Fredholm operators with weakly singular kernels. The proposed method is compared with existing methods available in the literature. Specifically, we highlighted its high accuracy and compared it with a recently developed hybrid numerical approach and finite difference methods. The efficiency and accuracy of the proposed method are demonstrated by solving several highly intermittent time-fractional nonlinear weakly singular integro-partial differential equations.

Contractivity of Möbius functions of operators

Let T be an injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers λ,μ for which (I+λT)(I+μT)−1 is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator T−1.When T=V, the Volterra operator on L2[0,1], this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those λ,μ for which (I+λV)(I+μV)−1 is a contraction. Taking T=Vn, we further deduce that (I+λVn)(I+μVn)−1 is never a contraction if n≥2 and λ≠μ.

Generalized Volterra Operators on Dales–Davie Algebras

Generalized Volterra Operators on Dales–Davie Algebras

Volterra and Composition Inner Derivations on the Fock–Sobolev Spaces

On the Fock–Sobolev spaces, we study the range of Volterra inner derivations and composition inner derivations. The Volterra inner derivation ranges in the ideal of compact operators if and only if the induced function g is a linear polynomial. The composition inner derivation ranges in the ideal of compact operators if and only if the induced function φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} is either identity or a contractive linear self-mapping of C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {C}$$\\end{document}. Moreover, we describe the compact intertwining relations for composition operators and Volterra operators between different Fock–Sobolev spaces. In this paper, our results are complement and in a sense extend some aspects of Calkin’s result (Ann Math 42:839–873, 1941) to the algebras of bounded linear operators on Fock–Sobolev spaces.

Analytic representations of backward shifts on weighted Bergman and Dirichlet spaces

The backward shifts (the adjoints of multiplication by z) on weighted Bergman spaces of the unit disk are important in operator theory. For example, Agler [1] shows their restrictions to invariant subspaces of vector-valued weighted Bergman spaces are model operators for hypercontractions. We present analytic representations of backward shifts on weighted Bergman spaces and Dirichlet type spaces. Since these analytic representations are not space specific, it is interesting to study these operators on Hilbert spaces of holomorprhic functions on the unit disk. We initiate this study by developing several properties of these operators. In particular, we obtain simple proofs and extend the invariant subspace theorem of shift plus Volterra operator on the Hardy space by Čučković and Paudyal [16]. Furthermore, we describe invariant subspaces of shift plus Volterra operator on the Bergman space by relating them to the invariant subspaces of the Dirichlet shift. Analytic representations of the tuple of backward shifts on weighted Bergman spaces of the unit ball are also discussed.

Investigation of conditions for preserving global solvability of operator equations by means of comparison systems in the form of functional-integral equations in $\mathbf{C}[0;T

Let $U$ be the set of admissible controls, $T>0$ and it be given a scale of Banach spaces $W[0;\tau]$, $\tau\in(0;T]$, such that the set of constrictions of functions from $W=W[0;T]$ to a closed segment $[0;\tau]$ coincides with $W[0;\tau]$; $F[\cdot;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. For the operator equation $x=F[x;u]$, $x\in W$, we introduce a comparison system in the form of functional-integral equation in the space $\mathbf{C}[0;T]$. We establish that, under some natural hypotheses on the operator $F$, the preservation of the global solvability of the comparison system pointed above is sufficient to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation. This fact itself is analogous to some results which were obtained by the author earlier. The central result of the paper consists in a set of signs for stable global solvability of functional-integral equations mentioned above which do not use hypotheses similar to local Lipschitz continuity of the right-hand side. As a pithy example of special interest, we consider a nonlinear nonstationary Navier–Stokes system in the space $\mathbb{R}^3$.

On the Real and Imaginary Parts of Powers of the Volterra Operator

On the Real and Imaginary Parts of Powers of the Volterra Operator

Debonding of an elastic layer with a cavity from a rigid substrate caused by rotation of a bonded rigid cylinder

Debonding of an elastic layer with a circular cylindrical cavity 0 ≤ ζ ≤ h , ρ ≥ 1 , from a rigid substrate under action of a rigid cylinder is the object of this study. The annular debonding zone 1 ≤ ρ < 1 + η is caused by rotation of a cylinder bonded to the cavity surface. The problem is reformulated as dual integral equations with Weber integral transforms kernels. A Volterra operator transforming a Weber transforms kernel into a Bessel function of the first kind and Hankel integral transforms allow us to reduce dual equations to a Fredholm integral equation of the second kind and then, by some transformation, to another Fredholm integral equation which is more suitable for approximate methods. As h ≥ 0 . 5 and 0 < η ≤ 1 , a highly accurate analytic approximate solution of the problem is suggested. The asymptotic solution of the problem is obtained as the width of debonding zone is very small while the thickness is not small. When the thickness is small, the Fredholm integral equations are computationally inefficient. A new method based on an operator transforming a Bessel function of the first kind into the kernel of Mehler–Fock integral transforms enabled us to convert one of the above-mentioned Fredholm equations into an equivalent Fredholm integral equation of the second kind that is effective for a small thickness. The asymptotic solution of the problem is obtained when both the layer thickness and the debonding zone width are small. Accurate mathematical methods, in particular investigations, and transformations of equations, developed in this study can be interesting to researchers employing dual integral equations technique in problems of mechanics and mathematical physics with mixed boundary conditions.

On fuzzy spectral radii for fuzzy bounded operators with application to fuzzy Volterra operator

On fuzzy spectral radii for fuzzy bounded operators with application to fuzzy Volterra operator

Further Properties of an Operator Commuting with an Injective Quasi-Nilpotent Operator

In (Aiena et al., Math. Proc. R. Irish Acad. 122A(2):101–116, 2022), it has been shown that a bounded linear operator T∈L(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T\\in L(X)$$\\end{document}, defined on an infinite-dimensional complex Banach space X, for which there exists an injective quasi-nilpotent operator that commutes with it, has a very special structure of the spectrum. In this paper, we show that we have much more: if a such quasi-nilpotent operator does exist, then some of the spectra of T originating from B-Fredholm theory coalesce. Further, the spectral mapping theorem holds for all the B-Weyl spectra. Finally, the generalized version of Weyl type theorems hold for T assuming that T is of polaroid type. Our results apply to the operators that belong to the commutant of Volterra operators.

On a convex combination of Lotka–Volterra operators

We consider a convex combination of two classes of Lotka–Volterra operators defined on 2-dimensional simplex. Earlier, the dynamics of a particular case of the considered operators has been investigated. However, its bijective property was not studied. In this paper, we are able to establish that such maps are homeomorphism of the simplex.

Some Operator Ideal Properties of Volterra Operators on Bergman and Bloch Spaces

Some Operator Ideal Properties of Volterra Operators on Bergman and Bloch Spaces

F-transform utility in the operational-matrix approach to the Volterra integral equation

The proposed approach, based on the F-transform, consists in constructing an operational matrix for the Volterra integral equation as a Hadamard product of two matrices, one of which refers to the Volterra operator, and the other to its kernel. As a result, the F-transformed form of the entire equation reduces to a system of linear equations with a triangular matrix. This makes the corresponding numerical method for solving the Volterra integral equation efficient and low computational. We provide all the necessary proofs of the supporting statements, including the convergence of the method, and estimate its computational complexity. Finally, we illustrate the proposed algorithm with test cases and compare their results with some recently published methods.

Generalized finite integration method with Volterra Operator for pricing multi-asset barrier option

We investigate in this paper the pricing of European-style barrier options under the Black–Scholes model. Based on the recently developed Generalized Finite Integration Method with Volterra operator (GFIM-V), we apply the Crank–Nicolson scheme to treat the time variable in the governing Black–Scholes equation for pricing multi-asset barrier options. For verification on the accuracy and efficiency of the proposed approach, we construct several numerical experiments for the solutions of multi-asset barrier option prices with various time step sizes and number of spatial nodal points. Comparisons with available exact solution and existing spectral convergent method indicate the advantages of the GFIM-V method in superior accuracy and unconditional stability.

Lyapunov functions and dynamics of infinite dimensional Volterra operators

The majority of research on quadratic stochastic operators (QSOs) was done on a finite-dimensional set of all probability distributions (also known as a simplex), thus it is intriguing to extend them for an infinite case. In particular, the infinite dimensional Volterra operators are discussed in the current paper. Due to the fact non-compactness of infinite dimensional simplex (unlike a finite case) makes the general study challenging, therefore, a sub-class of infinite dimensional Volterra operators is introduced. Furthermore, we construct Lyapunov functions which allow us to explore the dynamics of the introduced operators. Several examples are given with a full description of the limiting set.

On Singular Points of Linear Differential-Algebraic Equations with Perturbations in the Form of Integral Operators

The paper consideres linear systems of ordinary differential equations of arbitrary order with a matrix identically degenerate in the domain of definition at the highest derivative of the desired vector function and with loads in the form of Volterra and Fredholm integral operators. The initial value problems are formulated using projections onto admissible sets of initial vectors. Special attention is paid to systems having singular points on the interval of integration. The concept of a singular point is formalized. Their classification in the case of differential equations is given. A number of examples illustrating the theoretical results are presented.

Volterra operator norms : a brief survey

Abstract In this expository article, we discuss the evaluation and estimation of the operator norms of various functions of the Volterra operator.

A class of bijective Lotka–Volterra operators and its application

It is well known that any classical Lotka–Volterra (LV) operator (associated with quadratic stochastic operator) defined on the simplex is a homeomorphism. On the other hand, more general LV systems have important applications in the time evolution of conflicting species in biology. It is natural to study the bijectivity of such kind of LV operators. There is an example of a LV operator which is not injective. In this paper, we introduce a class of LV operators that are bijective. As an application of our result, the existence and uniqueness of solution of a class of Hammerstein integral equations is proved.