Optimal domain of Volterra operators in Korenblum spaces

Several notions of abstract Volterra operators on spaces of functions of one variable are well known. In this paper, we introduce various notions of abstract Volterra operators in spaces of functions of several variables. Some fixed point equations with such abstract Volterra operators are also studied. The basic ingredient in the theory of step by step contraction is the notion of G-contraction. The relevance of step by step contraction principle is illustrated by applications in the theory of Darboux-Ionescu problem.

Let V be the classical Volterra operator on L 2(0,1). Then the algebra generated (algebraically) by V and its adjoint is not only dense in the Banach space of all compact operators, but also in the Banach space of all Hilbert–Schmidt operators and as well in the space \({\mathcal{B}(L_2(0,1))}\) equipped with the weak operator topology. Moreover, the algebra generated by V 2 and its adjoint is dense in the Banach space of all trace class operators. We give an elementary proof that similar results are valid for polynomials in V without constant term. We also show that the commutant of any non-constant analytic function of V coincides with the commutant of V.

This chapter is devoted to operators of the type A=D+W, where D is a normal boundedly invertible operator in a separable Hilbert space H, and W has the following property: V:=D-1W is a Volterra operator in H. If, in addition, A has a maximal resolutions of the identity, then it is called a relatively P-triangular operator. Below we derive estimates for the resolvents of various relatively P-triangular operators and investigate spectrum perturbations of such operators.

Conditions are found which guarantee that a Volterra operator is equivalent to some power of the operator of indefinite integration in spaces of functions which are analytic in a disk.

49 - Inversion of bremsstrahlung spectra emitted by solar plasma

We apply our definition of Volterra operator on abstract spaces to some problems arising in metric spaces. In contrast to those known before, our definition requires only the existence of a σ-algebra on a metric space. Note that, being applied to such spaces, the new definition substantially extends the classes of operators of an evolutionary nature. It also allows one to relate different properties of the Volterra-type operators. In particular, the problem of quasi-nilpotentness studied traditionally in the Banach spaces only (since it requires equality to zero of the spectral radius of an operator) allows interpretation in complete locally convex spaces. Apparently, the question on preserving the Volterra property by a conjugate operator is posed for the first time. It should be mentioned that, generally speaking, the dual space to a Frechet (i.e., complete locally convex) space is not a Frechet space.

A generalized reproducing kernal Hilbert space (G-RKHS) of nonlinear Lipschitz operators is constructed for systems and control engineering applications. Specifically, the uniform topology is first introduced into the totality of one-parameter families of nonlinear Lipschitz operators to form a uniformly normed linear space, and then a generalized Bochner integral is introduced to define an operator-valued inner product structure and an induced norm for the space. It is shown that any closed and separable subspace of the resultant inner product space is a G-RKHS, which is a new mathematical structure. A generalized Fock space for the specific family of bounded nonlinear Volterra operators for multi-input/multi-output (MIMO) control systems can be constructed in the same manner. An application of the approach to a feedback design problem involving optimal disturbance rejection for general nonlinear MIMI control systems formulated in a Banach space setting is indicated. >

The Structure of Model Volterra Operators, Biorthogonal Expansions, and Interpolation in Regular de Branges Spaces

This paper considers the existence of $L^{p}$-solutions for a class of fractional integral equations involving abstract Volterra operators in a separable Banach space. Some applications for the existence of $L^{p}$-solutions for different classes of fractional differential equations are given.

On the linear equivalence of Volterra operators in Banach spaces

Let B be a dissipative Volterra operator in a separable Hilbert space \(H\) such that the resolvent \((I - zB)^{ - 1} \) has finite exponential type. A complete description is given of the operators B with the above properties, vectors \(g \in H\), and sequences \(\Lambda \) of complex numbers such that the family $$(I - \lambda _k B^2 )^{ - 1} , \lambda _k \in \Lambda ,$$ forms an unconditional basis in \(H\). Bibliography: 8 titles.

Some properties of volterra operators in analytic spaces

We present results on local solubility, extendability of solutions, and the existence of upper and lower solutions of equations with monotonic generalized Volterra operators in Banach function spaces. These results are analogous to the well-known theorems on the integral and differential inequality and can be used for estimating solutions of various functional-differential equations.

For operators acting in the Lebesgue space Lq(Π), 1 < q < ∞, an abstract analog of Bihari’s lemma is stated and proved. We show that it can be used to derive a uniform pointwise estimate of the increment of the solution of a controlled functional-operator equation in the Lebesgue space. The procedure of reducing controlled initial boundary-value problems to this equation is illustrated by the Goursat-Darboux problem.

A Banach space operator \(T\) is said to be weakly super convex-cyclic if there exists \(x \in X\) such that \(\{\lambda p(T )x : p\, \mathrm{convex \,polynomial}, \lambda \in \mathbb {C}\}\) is weakly dense in \(X\). The notion of convex-cyclicity was introduced recently by Rezaei in Linear Algebra Appl 438(11):4190–4203, (2013). We provide a simple argument, to show that many elements in the commutant of the Volterra operator acting on \(L^p_\mathbb {C}[0,1]\) spaces are not weakly super convex-cyclic.