Spectral analysis of finite convolution operators with matrix kernels

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)e^itzdt + e^izG(z), where G(z) is aB(N)-valued function which is bounded and analytic in a half plane y > . 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 > .This paper was written at the University of Virginia and is based on the author's doctoral dissertation 6], which was supervised by James Rovnyak.