Garske's inequality for an \(n\)-tuple of operators
For an arbitrary bounded linear operator \(T\) on a Hilbert space Garske's inequality \[ \sup_{\| x\|=1}\{\| Tx\|^ 2- |(Tx,x)|^ 2\}\geq R^ 2_ T\tag{1} \] is true. Here \(R_ T\) is the radius of the smallest disc containing the spectrum of \(\Gamma\). In the paper the following generalization of (1) is proved: Any \(n\)-tuple of commuting operators \(T=(T_ 1,T_ 2,\ldots,T_ n)\) satisfies the inequality (1), where \(R_ T\) is the radius of the smallest ball containing the Taylor spectrum of \(\Gamma\).
scientific article; zbMATH DE number 59721
scientific article