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Trace formulas and completely unitary invariants for some $$k$$-tuples of commuting operators. (English) Zbl 0827.47016
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, 367-380 (1995).
Summary: Let $$\mathbb{A}= (A_1, \dots, A_k)$$ be a $$k$$-tuple of commuting operators. Let $$A_1$$ be a pure subnormal operator with minimal normal extension $$N_1$$. Assume that $$\text{sp} (A_1) \setminus \text{sp} (N_1)$$ is a simply-connected domain with Jordan curve boundary satisfying certain smoothness conditions. Assume also that $$[A^*_i, A_j]\in {\mathcal L}^1$$, $$i,j=1, \dots, k$$. Then $$\mathbb{A}$$ is subnormal, and the set consisting of the $$\text{sp} (\mathbb{A})$$ and the function $$Q(\lambda)$$, $$\lambda\in \sigma_p (\mathbb{A}^*)^*$$ is a complete unitary invariant for $$\mathbb{A}$$, where $$Q(\lambda_1, \dots, \lambda_k)$$ is a parallel projection to $$\ker (A^*_1- \overline {\lambda}_1 I)\cap \{f: (A^*_j- \overline {\lambda}_j I)^\ell f=0$$, for some $$\ell>0$$, $$j=2, \dots, k\}$$.
For the entire collection see [Zbl 0819.00022].

##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc.