An Index Formula in Connection with Meromorphic Approximation

Let be a continuous matrix-valued function on the unit circle such that the st singular value of the Hankel operator with symbol is greater than the th singular value. In this case, it is well-known that has a unique superoptimal meromorphic approximant in ; that is, has at most poles in the unit disc (in the sense that the McMillan degree of in is at most ) and minimizes the essential suprema of singular values , with respect to the lexicographic ordering. For each , the essential supremum of is called the th superoptimal singular value of degree of . We prove that if has non-zero superoptimal singular values of degree , then the Toeplitz operator with symbol is Fredholm and has index where and denotes the Hankel operator with symbol . This result can in fact be extended from continuous matrix-valued functions to the wider class of -admissible matrix-valued functions, i.e. essentially bounded matrix-valued functions on for which the essential norm of the Hankel operator is strictly less than the smallest non-zero superoptimal singular value of degree of Phi.