Abstract
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
.