A distance formula for tuples of operators

Abstract: For a tuple of operators $\boldsymbol{A}= (A_1, \ldots, A_d)$, $\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})$ is defined as $\min\limits_{\boldsymbol{z} \in \mathbb C^d} |\boldsymbol{A-zI}|$ and $\text{var}x (\boldsymbol{A})$ as $|\boldsymbol{A} x|^2-\sum{j=1}^d {\big|}\langle x| A_j x\rangle{\big|}^2.$ For a tuple $\boldsymbol{A}$ of commuting normal operators, it is known that $$\text{dist}(\boldsymbol{A}, \mathbb C^d \boldsymbol{I})^2=\sup_{|x|=1}\text{var}_x (\boldsymbol{A}).$$ We give an expression for the maximal joint numerical range of a tuple of doubly commuting matrices. Consequently, we obtain that the above distance formula holds for tuples of doubly commuting matrices. We also discuss some general conditions on the tuples of operators for this formula to hold. As a result, we obtain that it holds for tuples of Toeplitz operators as well.