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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].
MSC:
 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