Determinants and a unified approach to estimating resolvents of operators in operator ideals of Riesz type \(\ell _ p\)
An operator ideal \({\mathfrak A}\) on the class of Banach spaces E is said to be of Riesz type \(\ell_ p\) \((0<p<\infty)\) if any \(T\in {\mathfrak A}(E)\) is a Riesz operator with absolutely p-summable eigenvalues. The main purpose of this paper is to give a unified approach to the problem of estimating resolvents of operators \(T\in {\mathfrak A}(E)\) that belong to an arbitrary operator ideal \({\mathfrak A}\) of Riesz type \(\ell_ p\). The method used is based on a representation of the resolvent of a power of T in the weak operator topology by characteristic determinants. The same method is used to derive estimates of Fredholm minors. This leads to generalizations and simplifications of various results on the growth of resolvents and Fredholm minors of operators that belong to some special operator classes (e.g. due to A. S. Markus, H. Köngi, and J. C. Engelbrecht and J. J. Grobler) and places these results in a natural setting. Moreover, this verifies a conjecture of H. König. Examples show that the estimates obtained are best possible. Finally, the resolvent estimates are applied to establish sufficient conditions for the completeness of principal vectors, thereby generalizing results of A. S. Markus, H. König, and the author.
Riesz operator with absolutely p-summable eigenvalues
representation of the resolvent of a power
scientific article; zbMATH DE number 3921322