##
**Joint quasitriangularity of essentially unitary pairs.**
*(English)*
Zbl 0938.47017

A celebrated result of Apostol, Foias and Voiculescu states that a Hilbert space operator \(T\) is quasitriangular if and only if the spectral picture of \(T\) contains no semi-Fredholm components with negative (Fredholm) index. In several variables no such description is yet available, although the paper under review, and several previous articles, have helped begin to unravel the (much more complicated) structure of jointly quasitriangular commuting \(n\)-tuples.

For \(T= (T_1,T_2)\) essentially normal, N. Salinas introduced [J. Oper. Theory 10, No. 1, 167-205 (1983; Zbl 0539.47011)] the subsemigroup \(\text{Ext}_{\text{qt}}(X)\) of \(\text{Ext}(X)\), where \(X\) is the essential spectrum of \(T\) and \(\text{Ext}(X)\) is built using the \(\text{BDF}\)-theory. In the present article, the authors focus on the case \(X\subseteq \mathbb{T}^2\) (essentially unitary pairs), mainly because of some substantial simplifications in the function-theoretic information, e.g., the description of \(\widehat X\), the polynomially convex hull of \(X\). In Theorem 2.3 it is shown that if \(X\) is the graph of a function \(h\in C(\mathbb{T})\), then either \(h\in A(\mathbb{D})\), \(X= \{(z,h(z)):z\in \mathbb{D}\}\), \(P(X)\cong A(\mathbb{D})\) and \(\text{Ext}_{\text{qt}}(X)\cong \mathbb{Z}_+\), or \(h\notin A(\mathbb{D})\), \(\widehat X=X\), \(P(X)= C(X)\) and \(\text{Ext}_{\text{qt}}(X)= 0\).

Next, the authors obtain similar results when \(X\) is a finite union of nice curves, particularly if the curves can be separated by analytic functions. As a direct consequence, they obtain a direct proof of G. Kaplan’s previous characterization of quasitriangular pairs with essential spectrum in the cyinder \(\mathbb{T}\times\mathbb{R}\).

For \(T= (T_1,T_2)\) essentially normal, N. Salinas introduced [J. Oper. Theory 10, No. 1, 167-205 (1983; Zbl 0539.47011)] the subsemigroup \(\text{Ext}_{\text{qt}}(X)\) of \(\text{Ext}(X)\), where \(X\) is the essential spectrum of \(T\) and \(\text{Ext}(X)\) is built using the \(\text{BDF}\)-theory. In the present article, the authors focus on the case \(X\subseteq \mathbb{T}^2\) (essentially unitary pairs), mainly because of some substantial simplifications in the function-theoretic information, e.g., the description of \(\widehat X\), the polynomially convex hull of \(X\). In Theorem 2.3 it is shown that if \(X\) is the graph of a function \(h\in C(\mathbb{T})\), then either \(h\in A(\mathbb{D})\), \(X= \{(z,h(z)):z\in \mathbb{D}\}\), \(P(X)\cong A(\mathbb{D})\) and \(\text{Ext}_{\text{qt}}(X)\cong \mathbb{Z}_+\), or \(h\notin A(\mathbb{D})\), \(\widehat X=X\), \(P(X)= C(X)\) and \(\text{Ext}_{\text{qt}}(X)= 0\).

Next, the authors obtain similar results when \(X\) is a finite union of nice curves, particularly if the curves can be separated by analytic functions. As a direct consequence, they obtain a direct proof of G. Kaplan’s previous characterization of quasitriangular pairs with essential spectrum in the cyinder \(\mathbb{T}\times\mathbb{R}\).

Reviewer: Raul E.Curto (MR 94j:47029)

### MSC:

47A66 | Quasitriangular and nonquasitriangular, quasidiagonal and nonquasidiagonal linear operators |

32E20 | Polynomial convexity, rational convexity, meromorphic convexity in several complex variables |

47A10 | Spectrum, resolvent |

47A13 | Several-variable operator theory (spectral, Fredholm, etc.) |

47B20 | Subnormal operators, hyponormal operators, etc. |