Linear rank and corank preserving maps on \({\mathcal B}(H)\) and an application to \(^*\)-semigroup isomorphisms of operator ideals (Q1808939)

The article presents some results in the sphere of linear operators in Hilbert space. Let \(B(H)\) be the algebra of all bounded linear operators on the Hilbert space \(H\). Let \(\phi\) be a linear map \(\phi:B(H)\to B(H)\) and \(\phi\) be preserving rank or corank, or nonincreasing rank of all elements of \(B(H)\). The structure of the map \(\phi\) is also shown in the work. Besides, the authors prove theorems on isomorphisms of operator ideals in \(B(H)\). For finite-dimensional results see \textit{L. B. Beasley} [Linear Algebra Appl. 107, 161-167 (1988; Zbl 0651.15004)] and \textit{R. Loewy} [Linear Multilinear Algebra 34, No. 1, 21-32 (1993; Zbl 0801.15002)], for infinite-dimensional results rank \(=1\) see \textit{J. Hou} [Sci. China, Ser. A 32, No. 8, 929-940 (1989; Zbl 0686.47030)] and \textit{M. Omladič} and \textit{P. Šemrl} [Linear Algebra Appl. 182, 239-256 (1993; Zbl 0803.47026)].