A Geometric Approach to Positive Definite Sequences

Abstract

In this chapter we present a geometric approach for studying positive definite sequences. First, we will obtain the Naimark dilation theorem which is the fundamental result of this chapter. Then we will use this theorem to show that there is a one to one correspondence between the set of all positive definite sequences on H, and the set of all choice sequences initiated on H. We also solve the positive definite Caratheodory interpolation problem in the operator setting, and present a geometric proof for many of the results in Chapter II. Then a recursive inverse scattering algorithm to compute the choice sequence from the positive definite sequence is presented. Further, we use a special matrix representation of the Naimark dilation to demonstrate how this choice sequence occurs in the Levinson algorithm for block Toeplitz matrices. This leads to another inverse scattering algorithm to compute the choice sequence. Also shown is how the Naimark dilation occurs in an operator version of the marine seismology and layered medium models discussed in Chapter III. Finally, we will show that there is a one to one correspondence through the Cayley transform, between the set of all positive definite sequences on H, and the set of all contractive analytic functions in H∞(H, H). From this it will follow that the positive definite operator valued Caratheodory interpalation problem is equivalent to the usual operator valued Caratheodory interpalation problem involving n by n contractive analytic Toeplitz matrices.