Properties of total sequences of operators and some of their applications
Let X and Y be normed linear spaces. A sequence of linear operators \(\{A_ n\}\), each \(A_ n\) acting on X into Y, is said to be total if \(\cap^{\infty}_{n=1}Ker A_ n=\{0\}.\) A sequence of bounded linear operators \(\{B_ n\}\) \((n=1,2,...)\) is said to be dually total if the sequence of dual operators \(\{B_ n^*\}\) is total. In this paper the author characterized total and dually total sequences in terms of the fundamentalness of certain linear sets and formulated some necessary conditions for the totalness, dual totalness of sequences of operators. Using these results the author obtained necessary conditions for the linear regularizability of the inverse operator \(A^{-1}\), where \(A\in B_ 0(X,Y)(B_ 0(X,Y)\) is the set of all bounded injective operators). And also the author gives two explicit examples of \(B_ 0(X,Y)\) whose inverse operators are not linearly regularizable, namely that \(B_ 0(C^*[0,1],L_{\infty}[0,1])\) and \(B_ 0(\ell_ 2(\Gamma),\ell_{\infty}(\Gamma)),\) \(\Gamma\) is nonseparable.
total and dually total sequences in terms of the fundamentalness of certain linear sets
linear regularizability of the inverse operator
scientific article; zbMATH DE number 3909163