zbMATH — the first resource for mathematics

Spectral picture and index invariants of commuting \(n\)-tuples of operators. (English) Zbl 0870.47005
Curto, Raúl E. (ed.) et al., Multivariable operator theory. A joint summer research conference on multivariable operator theory, July 10-18, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 185, 219-236 (1995).
Let \(T=(T_1;T_2;\dots;T_n)\) be an essentially commuting \(n\)-tuple of bounded operators acting in the Banach space \(X\). In the present paper the \(n\)-tuples of essentially commuting bounded operators are considered from the topological point of view. Namely, for essentially commuting \(n\)-tuples \(T\) one obtains the index class \(k(T)\) as an element of the \(K^0\)-group of the complement of the Taylor’s essential spectrum \(\sigma_e(T)\). In the case of the commuting \(n\)-tuples \(k(T)\) is completely determined by a certain complex-analytic object and the corresponding invariant in cohomology \(ch(T):= ch(k(T))\) takes values in the Hodge cohomologies. In the case of the essentially commuting \(n\)-tuples one obtains the invariant \(K(T)\) as an element of the Grothendieck group of the category of coherent sheves on \(\mathbb{C}^n\setminus \sigma_e(T)\).
The first part of the paper concerns the definition and the functorial properties of the invariants \(k(T)\), \(ch(T)\), \(K(T)\). The second part concerns some applications:
i) a very short proof of the Boutet de Monvel formula for the index of Toeplitz operators and a generalization of this theorem are given;
ii) the problem of the existence of a commuting compact perturbation of the given essentially commuting \(n\)-tuple.
For the entire collection see [Zbl 0819.00022].
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
46M20 Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.)
47A53 (Semi-) Fredholm operators; index theories
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47A10 Spectrum, resolvent