Perturbation analysis of Moore-Penrose quasi-linear projection generalized inverse of closed linear operators in Banach spaces (Q296173)

For real vector spaces \(U,V\), a mapping \(T:U \rightarrow V\) is said to be \textit{quasi-additive} relative to \(L \subseteq U\) if \(T(x+z)=T(x)+T(z)\) whenever \(x \in U\) and \(z \in L\). A mapping \(S:V \rightarrow V\) is said to be a \textit{quasi-linear projector}, if \(S\) is homogeneous (meaning that \(S(\lambda x) = \lambda S(x)\) for each \(\lambda \in \mathbb{R}\) and \(x \in V\)), idempotent and quasi-additive relative to the range of \(S\). Let \(X,Y\) be normed linear spaces and \(T\) be a linear operator from \(X\) into \(Y\). Suppose that there exist two bounded quasi-linear projectors \(S_1\) and \(S_2\) from \(X,Y\) onto the closures of \(N(T), R(T)\), respectively. A quasi-additive homogeneous operator \(A\) from \(Y\) into \(X\) is called the \textit{Moore-Penrose bounded quasi-linear projection generalized inverse} of \(T\) if \(TAT=T\) on \(D(T)\), \(ATA=A\) on \(D(A),~AT=I-S_1\) on \(D(T)\) and \(TA=S_2\) on \(D(A)\) (\(D(\cdot)\) denoting the domain). When such an \(A\) exists, it is denoted by \(T^h\). The main result of the authors is the following: Let \(X,Y\) be Banach spaces and \(T\) be a closed densely defined linear operator from \(X\) into \(Y\) with the range space of \(T\) being closed. Suppose that the Moore-Penrose bounded quasi-linear projection generalized inverse \(A\) of \(T\) exists. Let \(E\) be a linear operator from \(X\) into \(Y\) such that \(D(E)=D(T), N(T) \subseteq N(E), R(E) \subseteq R(T)\) and satisfy the inequality \(\| Ex \| \leq a \| x \| + b \| Tx \|\) for every \(x \in D(T)\) for some nonnegative numbers \(a, b\). Let \(a \| A \| + b \| S_2 \| <1,\) where \(S_2\) is defined as above. Set \(W=T+E\), the perturbation of \(T\) by \(E\). Then \(W\) is a closed operator, \(R(W)=R(T)\), \(N(W)=N(T)\) and the Moore-Penrose bounded quasi-linear projection generalized inverse of \(W\) exists. An explicit formula for \(W^h\) is derived and bounds on the norms of \(W^h\) and \(W^h-T^h\) are obtained.