Distance between commuting tuples of normal operators (Q1270254)

An \(m\)-tuple \(\mathbf{A} = (A_1, \dots, A_m)\) of (Hilbert space bounded linear) operators is identified with an operator to the direct sum. Then its norm is defined naturally as \(\|\mathbf{A}\|= \|\sum_{j=1}^mA_j^*A_j\|^{1/2}\). For an \(m\)-tuple of scalars \(\underline{\lambda}= (\lambda_1, \ldots, \lambda_m)\) this norm is just the Euclidean length. It is proved that for two commuting \(m\)-tuples of normal operators \(\mathbf{A} = (A_1, \dots, A_m), \text\textbf{B} = (B_1, \dots, B_m)\) \[ \|\mathbf{A} - \text\textbf{B}\|\leq \sqrt{2}\max\{ \|\text\textbf{\(\lambda\)}- \text\textbf{\(\mu\)}\|; \;\text\textbf{\(\lambda\)} \in \sigma(\text\textbf{A}), \text\textbf{\(\mu\)} \in \sigma(\text\textbf{B})\} \] where \(\sigma(\mathbf{A})\) denotes the joint spectrum of \(\mathbf{A}\).