On the Fredholm and Weyl spectra of several commuting operators (Q2464661)

The Taylor joint spectrum \(\sigma(T)\) of an \(n\)-tuple \(T=(T_1,\dots,T_n)\) of mutually commuting bounded linear operators on a Banach space \(X\) consists of all \(n\)-tuples \(\lambda=(\lambda_1,\dots,\lambda_n)\in{\mathbb C}^n\) for which the Koszul complex \(K_*(T-\lambda,X)\) of the operators \(T_1-\lambda_1,\dots,T_n-\lambda_n\) is not exact. The Fredholm spectrum \(\sigma_F(T)\) is defined to be the set of all \(\lambda\in\sigma(T)\) for which all the homology spaces \(H_i(T,\lambda)=H_i(K_*(T-\lambda,X))\) are finite-dimensional. The difference \(\sigma(T)\setminus\sigma_F(T)\) is called essential spectrum and is denoted by \(\sigma_e(T)\). The paper continues the investigations of the structure of the Fredholm spectrum of commuting \(n\)-tuples of operators started by the author in [J.~Oper.\ Theory 21, No.\,2, 219--253 (1989; Zbl 0704.46047)]. Any finite Fredholm complex of Banach spaces with differentials holomorphically depending on the parameters is locally holomorphically quasi-isomorphic to a holomorphic complex of finite-dimensional spaces. Therefore, the homology sheaves \({\mathcal H}_i(T)\) of the complex of germs of holomorphic functions with values in \(K_*(T-\lambda,X)\) are coherent on \({\mathbb C}\setminus\sigma_e(T)\). For a point \(\lambda^0\in\sigma_F(T)\), the stalk of the homology sheaf \({\mathcal H}_i(T)_{\lambda^0}\) can be considered as a module over the Noetherian local ring \({\mathcal O}_{\lambda^0}\) of germs of holomorphic functions at this point. The Fredholm spectrum \(\sigma_F(T)\) is a complex-analytic subspace of \({\mathbb C}^n\setminus\sigma_e(T)\). Its dimension near \(\lambda^0\), being well-defined, is an integer not exceeding \(n\). There are many results in the case when the Fredholm spectrum is is of maximal dimension, especially for sub- and factor-modules of spaces of analytic functions. The paper deals mainly with the case when this dimension is less than \(n\). To every coherent sheaf, or finitely generated \({\mathcal O}_{\lambda^0}\)-module, one can attach an element of the cycle group of \({\mathcal O}_{\lambda^0}\), i.e., a formal sum of prime ideals of \({\mathcal O}_{\lambda^0}\). Taking the alternated sum of cycles of the modules \({\mathcal H}_i(T)_{\lambda^0}\) for \(i=0,1,\dots,n\), one obtains the cycle of the Koszul complex of the \(n\)-tuple \(T-\lambda^0\). In this way, a set of integers characterizing the homology sheaves of the Koszul complex of \(T\) is defined. In the case of a single operator \(T\), the corresponding invariant is the index of \(T-\lambda^0\). The paper establishes some connections between these algebraic characteristics of the homology sheaves and the action of the operators \(T_1,\dots, T_n\) on \(X\). The author also introduces a new notion of the Weyl joint spectrum and proves some of its properties.