Joint spectra of doubly commuting \(n\)-tuples of operators and their Aluthge transforms. (Q1841607)

For a separable infinite-dimensional complex Hilbert space \(H\), let \(B(H)\) denote the algebra of all bounded linear operators on \(H\). If \(A\in B(H)\), it has a unique polar decomposition \(A= U| A|\), where \(| A|= (A^* A)^{{1\over 2}}\) and \(U\) is a partial isometry. Associated with \(A\) there is useful related operator \(\widetilde A=| A|^{{1\over 2}} U| A|^{{1\over 2}}\), called the Aluthge transform of \(A\) [see \textit{A. Aluthge}, Integral Equations Oper. Theory 13, 307--315 (1990; Zbl 0718.47015)]. Let \(\mathbb{A}= (A_1,\dots, A_n)\) denote a commuting \(n\)-tuple of operators and denote \(\mathbb{A}^*= (A^*_1,\dots, A^*_n)\), \(\widetilde{\mathbb{A}}= (\widetilde A_1,\dots,\widetilde A_n)\) and \((\widetilde{\mathbb{A}})^*= ((\widetilde A_1)^*,\dots, (\widetilde A_n)^*)\). If \(A_iA_j= A_jA_i\) and \(A^*_i A_j= A_j A^*_i\) for every \(i\neq j\), then \(\mathbb{A}\) is said to be a doubly commuting \(n\)-tuple. For a commuting \(n\)-typle \(\mathbb{A}\), let \(\sigma_p(\mathbb{A})\) [resp. \(\sigma_{ap}(\mathbb{A})\), \(\sigma^l_e(\mathbb{A})\), \(\sigma^r_e(\mathbb{A})]\) denote the joint point spectrum [resp., the joint approximate point spectrum, left joint essential spectrum, right joint essential spectrum]. \textit{I.~B.~Jung}, \textit{E.~Ko} and \textit{C.~Percy} [Integral Equations Oper. Theory 37, 437--448 (2000; Zbl 0996.47008)] proved that for an arbitrary \(A\in B(H)\) various spectra of \(A\) coincide with those of \(\widetilde A\). The authors extend these results to a doubly commuting \(n\)-tuple \(\mathbb{A}= (A_1,\dots, A_n)\) of operators in \(B(H)\). In fact, they show that \(\sigma_p(\mathbb{A})= \sigma_p(\widetilde{\mathbb{A}})\), \(\sigma_{ap}(\mathbb{A})= \sigma_{ap}(\widetilde{\mathbb{A}})\), \(\sigma_p(\mathbb{A}^*)\setminus[0]= \sigma_p((\widetilde{\mathbb{A}})^*)\setminus[0]\), \(\sigma_{ap}(\mathbb{A}^*)\setminus[0]= \sigma_{ap}((\widetilde{\mathbb{A}})^*)\setminus[0]\), \(\sigma^l_e(\mathbb{A})= \sigma^l_e(\widetilde{\mathbb{A}})\), and \(\sigma^r_e(\mathbb{A})\setminus[0]= \sigma^r_e(\widetilde{\mathbb{A}})\setminus[0]\). Moreover, if each \(A_i\) is such that the partial isometry in its polar decomposition is unitary, then they prove that \(\sigma_p(\widetilde{\mathbb{A}}^*)= \sigma_p((\widetilde{\mathbb{A}})^*)\), \(\sigma_{ap}(\widetilde{\mathbb{A}}^*)= \sigma_{ap}((\widetilde{\mathbb{A}})^*)\), and \(\sigma^r_e(\mathbb{A})= \sigma^r_e(\widetilde{\mathbb{A}})\).