Computing the Fredholm Index of Toeplitz Operators with Continuous
Symbols
Joint with Paul McGuire
To appear in Proc. Amer. Math. Soc.
dvi
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If S is a subnormal operator on a Hilbert space H with minimal normal
extension N acting on the space K and f is a continuous function on the
spectrum of N, then one can use the functional calculus for normal
operator to create the operator f(N) on K. Compressing f(N) to H
gives the Toeplitz operator $S_f$ with symbol f constructed from the
subnormal operator S. $S_f$ can also be defined as a limit of
polynomials in S and S*, thus $S_f$ can truly be thought as
"constructed from S".
In 1982 Olin & Thomson showed that if S is an essentially normal
subnormal operator (so its self-commutator (S*S-SS*) is compact), then
the essential spectrum of $S_f$ may be computed as $f(\sigma_e(S))$
that is simply the range of f on the essential spectrum of S.
In this paper we show how to compute the Fredholm index of $S_f$ in
terms of limits of sums of winding numbers when S is essentially
normal. The critical step is to compute the index of $S_f$ when S
is multiplication by z on the Hardy space of an arbitrary bounded
region. This uses Brown-Douglas-Fillmore Theory. Then again
using BDF-Theory we use the Hardy space case to prove the general case
for an arbitrary essentially normal subnormal operator S.
We also raise questions about the spectral pictures of Toeplitz
operators (the spectral picture consists of the essential spectrum and
the values of the index function off the essential spectrum). We
show that a Toeplitz operator can have any prescribed spectral
picture. We ask if this is still true if S is also required to be
irreducible. And even harder, if S is required to be irreducible
and f is required to be one-to-one on the essential spectrum.
This latter question is related to finding subnormal generators of
C*-algebras.