To appear in Proc. Amer. Math. Soc.

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.