Spectral analysis of finite convolution operators with matrix kernels

Abstract

Let N be a finite dimensional complex Hilbert space. A finite convolution operator on the vector function space L N 2 (0,1) is an operator T of the form (Tf) (x) = ∫ 0 x k (x−t)f(t)dt, where k(t) is a norm integrableB(N)-valued function on [0,1]. A symbol for T is any function A(z) of the form A(z) = ∫ 0 1 k(t)eitzdt + eizG(z), where G(z) is aB(N)-valued function which is bounded and analytic in a half plane y > η. It is shown that under suitable restrictions two finite convolution operators are similar if their symbols are asymptotically close as z → ∞ in a half plane y > η .