Relative numerical ranges
Let \(H\) be a separable complex Hilbert space and let \(B(H)\) be the Banach algebra of all bounded linear operators on \(H\). The numerical range of \(S\in B(H)\) is the set \(W(S)=\left\{(Sx,x):\; ||x||\; =1\right\}\). It is known that the spectrum \(\sigma (S)\) of \(S\) is contained in \(W(S)\). In this paper, the authors define a more general set, the numerical range of \(S\) relative to some operator \(T\in B(H)\) at a point \(r\in \sigma (|T|)\). Namely, this is the set \[ W_{T}^{\, r} (S)=\left\{\lambda \in {\mathbb C}:\exists (x_{n} ),\; ||x_{n} ||\; =1,\; \mathop{\lim }\limits_{n\to \infty } ||\; |T|x_{n} -r{\kern 1pt} x_{n} ||\; =0\; \; \text{and}\; \; \mathop{\lim }\limits_{n\to \infty } (S{\kern 1pt} x_{n} ,x_{n} )=\lambda \right\}. \] The authors prove various properties of this numerical range. In particular, \(W_{T}^{\, r} (S)\) is a non-empty convex subset of the closure of \(W(S)\). It is shown that the presence of 0 in \(W_{T}^{\, r} (S*T)\) gives a lower bound for the distance from \(T\) to the linear space spanned by \(S\).
scientific article; zbMATH DE number 6493726
scientific article