An Index Formula for an n-Tuple of Shifts on the Polydisk

Abstract

Let (Mz1, ... , MZn ) be an n-tuple of shift operators on the polydisk 12(Zn); we compress it to a variety of subspaces of 12(Zn) that are combinatorially constructed. The main result is a multivariate Fredholm index formula, which links the indices of the n-tuples to their combinatorial data in the definitions of the subspaces. In [11], Taylor, using homological algebraic methods, introduced a natural spectral theory for n-tuples of commuting operators acting on Banach spaces. Fredholm theory and index theory are developed with respect to this spectral theory [2, 3, 8]. While this several variable spectral theory is a natural generalization of the classical single operator theory (when n = 1), the techniques of computing these spectra are so different from or, much more complicated than, that of single operators. The problem of deciding the spectral pictures of various classes of n-tuples of operators is very essential in multivariate operator theory and has deep applications to many areas of analysis, such as function theory on domains in Cn . Many examples have been investigated through many different techniques [4, 5, 7, 9, 10]. In this note, we use the homological method to compute the spectra and Fredholm indices of a new, interesting class of n-tuples of operators. We will give an index formula for these n-tuples in terms of combinatorial data in their definition. An interesting consequence of this is that some combinatorial property of the operator forces the index to be nonzero. To be more precise, we denote by T = (T1, . .. , Tn) an n-tuple of commuting operators on the Hilbert space H and by K(T, z) = {Kq (T, z), 0q } the Koszul complex of T at z E Cn, where Kq (T, z) is the space of q-cochains and aq is the differential map at degree q. We refer the reader to Taylor's original papers [1 1, 12] for a more detailed account of these concepts. Recall that a point z in Cn is said to be in the Taylor spectrum a(T), if the Koszul Received by the editors October 6, 1992. 1991 Mathematics Subject Classification. Primary 47D25. Research supported in part by a grant from NSF, DMS 90-02969. (? 1994 American Mathematical Society 0002-9939/94 $1.00 + $.25 per page