Invertibility of Multidimensional Integral Operators with Bihomogeneous Kernels

Mathematical Notes -     

On the Pseudospectra of Multidimensional Integral Operators with Homogeneous Kernels of Degree -n

We study the limit behavior of the spectral characteristics of truncated multidimensional integral operators whose kernels are homogeneous of degree –n and invariant under the rotation group SO(n). We prove that the limit of the -pseudospectra of the truncated operators K as 0 is equal to the union of the -pseudospectra of the original operator K and the transposed operator .     

An algebra generated by multiplicative discrete convolution operators

We consider a Banach algebra generated by multiplicative discrete convolution operators. We construct a symbolic calculus for this algebra and in terms of this calculus we describe criteria for the Noetherian property of operators and obtain a formula for their index.     

Compactness of Some Operators of Convolution Type in Generalized Morrey Spaces

Sufficient conditions for the compactness in generalized Morrey spaces of the composition of a convolution operator and the operator of multiplication by an essentially bounded function are obtained. Very weak conditions on the function are also obtained under which the commutator of the operator of multiplication by such a function and a convolution operator is compact. The compactness of convolution operators in domains of cone type is investigated.     

Multidimensional integral operators with homogeneous-difference kernels

We consider multidimensional integral operators with homogeneous-difference kernels acting in L _p -spaces. For such operators, we prove a boundedness theorem and establish an invertibility criterion.     

Multidimensional integral operators with bihomogeneous kernels: A projection method and pseudospectra

We consider multidimensional integral operators with bihomogeneous and rotation invariant kernels. For such operators we obtain a criterion for applicability of a projection method in the scalar and matrix cases and describe the limit behavior of the ε-pseudospectra of the truncated operators $A_{\tau _1 , \tau _2 } $ as τ ₁ → 0 and τ ₂ → 0.     

On the Algebra of Pair Integral Operators with Homogeneous Kernels

In this paper, we study the Banach algebra generated by multidimensional pair integral operators with homogeneous kernels. We describe necessary and sufficient conditions for operators from the algebra to be Fredholm and present a formula for calculating the index.     

On an algebra of multidimensional integral operators with homogeneous-difference kernels

A Banach algebra generated by multidimensional integral operators with homogeneous-difference kernels is considered. A symbolic calculus is constructed for this algebra and, in terms of this calculus, necessary and sufficient conditions for the invertibility of an operator are obtained.