Generalizations of a theorem of S. M. Nikol'skij for normally solvable operators in Hilbert spaces (Q1595638)

Let \(H_1\), \(H_2\) be Hilbert spaces and let \(L:H_1\to H_2\) be a bounded linear operator. There are given necessary and sufficient conditions for \(L= B+S\) with a bounded inverse (left, right inverse) of \(B\) and a properly chosen projector \(B^{-1}S\) to be normally solvable, in particular to have the kernel \(N(L)\) isomorphic to \(N(L^*)\). This last case corresponds to the classical Nikol'skij theorem in Banach spaces from 1943. Remark of the reviewer: An algebraic generalization of the above results is the following. Let \(X\), \(Y\) be linear spaces (without any topology) and let \(T\in L(X,Y)\). Then \(\dim N(T)= (\geq,\leq)\) codim range of \(T\) if and only if \(T= S+K\), where \(S\) is invertible (left, right invertible) and \(K\) is finite-dimensional [cf. the reviewer and \textit{S. Rolewicz}, ``Equations in linear spaces'', Monografie Matem. 47, Warszawa (1968; Zbl 0181.40501)].