Numerical range of two operators in semi-inner product spaces
The authors define, for any two operators \(T\) and \(A\) on a complex semi-inner product space \((X,[.,.])\) with domains \(D(T)\) and \(D(A)\), their numerical range \(W(T,A)\equiv \{[Tx, Ax]:\| Ax\|= 1\), \(x\in D(T)\cap D(A)\}\), the spectrum \(\sigma(T,A)\equiv \{\lambda\in\mathbb{C}: T-\lambda\) is not invertible\}, and other familiar spectral subsets of the latter. Then they show that \(\sigma(T,A)\) is contained in the closure of \(W(T,A)\), thus generalizing the classical result for a single operator \(T\). The authors proceed to define such sets associated with two nonlinear operators \(T\) and \(A\), and prove an analogous containment relation between the approximate point spectrum and the closure of the numerical range of \(T\) and \(A\). Finally, they establish an approximation method for solving the operator equation \((A-T)y= Ax\) for nonlinear \(T\) and \(A\) on a complex Banach space.
scientific article; zbMATH DE number 6034251
scientific article