Integral Equations and Operator Theory Vol. i0 (1987) 0378-620X/87/050707-1451.50+0.20/0 (c) 1987 Birkhguser Verlag, Basel ELEMENTARY OPERATORS WITH H -SYMBOLS Rafll E. Curto and Lawrence A. Fialkow ~ Let T be a c.n.u, contraction on a Hilbert space ~ and let u = (Ul,~ be an n-tuple of H~(T)-functions. We calcu- late various joint spectra of u(T) and apply the results to study the spectral and index theories of elementary operators associated with n-tuples of the above type. 1. INTRODUCTION Let T be a completely nonunitary (c.n.u.) contrac- tion acting on a Hilbert space ~, and let u = (Ul,---,Un) be an n-tuple os H~(~)-s (Y is the unit circle). LeZ u(T) := (Ul(T),'~ where each ui(T ) is given by the Sz. Nagy-Foia~ functional calculus. When n = 1, C. Foia~ and W. Mlak studied in [O] the spectrum of u(T), a(u(T)), in terms of the spectrum of T. They proved that aCu(W)) = UCaext(T)) , where aext(T ) is a certain compact nonempty subset of the maxi- mal ideal space of H~(T), M and ^ stands for the H~(V) ' Gelfand transform. Foia~ and Mlak then looked for geometric descriptions of U(aext(T}), and, in particular, they proved that A UC~extCT)) = uCoCT)) whenever u can be continuously extended to a(T) Q Y. Both authors have been partially supported by NSF grants.