arXiv subjects
Publications and source records attributed to Igor M. Novitskii.
In this paper, we approximate to continuous bi-Carleman kernels vanishing at infinity by sequences of their subkernels of Hilbert-Schmidt type and try to construct the resolvent kernels for these kernels as limits of sequences of the resolvent kernels for the approximating subkernels
We prove that, at regular values lying in a strong convergence region, the resolvent kernels for a continuous bi-Carleman kernel vanishing at infinity can be expressed as uniform limits of sequences of resolvent kernels for its approximating subkernels of Hilbert-Schmidt type
In this paper, we study an infinite system of Fredholm series of polynomials in $λ$, formed, in the classical way, for a continuous Hilbert-Schmidt kernel on $\mathbb{R}\times\mathbb{R}$ of the form $\boldsymbol{H}(s,t)-λ\boldsymbol{S}(s,t)$, where $λ$ is a complex parameter. We prove a convergence of these series in the complex plane with respect to sup-norms of various spaces of continuous functions vanishing at infinity. The convergence results enable us to solve explicitly an integral equation of the second kind in $L^2(\mathbb{R})$, whose kernel is of the above form, by mimicking the classical Fredholm-determinant method.
With the aim of applications to solving general integral equations, we introduce and study in this paper a special class of bi-Carleman kernels on $\mathbb{R}\times\mathbb{R}$, called $K^\infty$ kernels of Mercer type, whose property of being infinitely smooth is stable under passage to certain left and right multiples of their associated integral operators. An expansion theorem in absolutely and uniformly convergent bilinear series concerning kernels of this class is proved extending to a general non-Hermitian setting both Mercer's and Kadota's Expansion Theorems for positive definite kernels. Another theorem proved in this paper identifies families of those bounded operators on a separable Hilbert space $\mathcal{H}$ that can be simultaneously transformed by the same unitary equivalence transformation into bi-Carleman integral operators on $L^2(\mathbb{R})$, whose kernels are $K^\infty$ kernels of Mercer type; its singleton version implies in particular that any bi-integral operator is unitarily equivalent to an integral operator with such a kernel.
In this paper, we reduce the general linear integral equation of the third kind in $L^2(Y,μ)$, with largely arbitrary kernel and coefficient, to an equivalent integral equation either of the second kind or of the first kind in $L^2(\mathbb{R})$, with the kernel being the linear pencil of bounded infinitely differentiable bi-Carleman kernels expandable in absolutely and uniformly convergent bilinear series. The reduction is done by using unitary equivalence transformations.
In this paper, we describe families of those bounded linear operators on a separable Hilbert space that are simultaneously unitarily equivalent to integral operators on $L_2(R)$ with bounded and arbitrarily smooth Carleman kernels. The main result is a qualitative sharpening of an earlier result of [7].
In this paper, we characterize all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral bi-Carleman operator in $L_2(R)$ with bounded and arbitrarily smooth kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators. The main result is a qualitative sharpening of an earlier result of [5].
In this paper, we characterize the families of those bounded linear operators on a separable Hilbert space which are simultaneously unitarily equivalent to integral bi-Carleman operators on $L_2(R)$ having arbitrarily smooth kernels of Mercer type. The main result is a qualitative sharpening of an earlier result of [7].
In this paper, we give a characterization of all closed linear operators in a separable Hilbert space which are unitarily equivalent to an integral operator in $L_2(R)$ with bounded and arbitrarily smooth Carleman kernel on $R^2$. In addition, we give an explicit construction of corresponding unitary operators.