Operator balls
Let \(H_ 1,H_ 2\) be Hilbert spaces and let \({\mathcal K}({\mathcal K}^ 0)\) denote the set of all bounded linear operators from \(H_ 1\) to \(H_ 2\) with norm \(\leq 1\) (resp. \(<1\)). For operators \(A,B,C\), the set \({\mathcal K}(C;A,B)=C+A{\mathcal K}B\) \(({\mathcal K}^ 0(C;A,B)=C+A{\mathcal K}^ 0B)\) is called the closed (resp. open) operator ball of the title. \(C\) is the `centre', \(A\), \(B\) are `radii'. Note that \({\mathcal K}(C;A,B)\) is the closure of \({\mathcal K}^ 0(C;A,B)\) in the weak, strong, and operator topologies but \({\mathcal K}^ 0(C;A,B)\) is not (relatively) open in any of the usual topologies. Many results are given including necessary and sufficient conditions for inclusions between operator balls to hold. For example, either an open ball \({\mathcal K}^ 0(C;A,B)\) is contained in \({\mathcal K}^ 0(C_ 1;A_ 1,B_ 1)\) or the entire closed ball \({\mathcal K}(C;A,B)\) lies on the boundary of \({\mathcal K}(C_ 1;A_ 1,B_ 1)\). Connections with the study of linear fractional transformations are promised for a later paper.
scientific article; zbMATH DE number 7069
scientific article