Asymptotic behavior of pseudo-resolvents on some Grothendieck spaces
Let B(X) be the set of all bounded linear operators on a Banach space; a family \(\{J_ i\); \(i\in I\}\) of operators in B(X) is called a pseudo- resolvent on \(I\subset {\mathbb{C}}\) if \[ J_ i-J_ k=(k-i)J_ iJ_ k,\quad \forall i,k\in I. \] X is called a Grothendieck space if every \(w^*\)-convergent sequence in the dual space \(X^*\) is weakly convergent; X is called to have the Dunford-Pettis property if \(<x_ n,x^*_ n>\to 0\) whenever \((x_ n)\subset X\) tends weakly to 0 and \((x^*_ n)\subset X^*\) tends weakly to 0. In this paper, the author proves that: If X is a Grothendieck space, \(\lambda J_{\lambda}\) is convergent in the strong operator topology as \(\lambda\) \(\to 0\), if and only if \(\| \lambda J_{\lambda}\| =O(1)\) as \[ \overline{R(\lambda J^*_{\lambda}-I^*)}=w^*- cl(R(\lambda J^*_{\lambda}-I^*). \] The proof is a direct consequence of a theorem of Yosida. If, in addition, X has the Dunford Pettis property, the strong convergence of \(\lambda J_{\lambda}\) is equivalent to the uniform convergence.
scientific article; zbMATH DE number 4075871