×

zbMATH — the first resource for mathematics

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.
PDF BibTeX XML Cite