Abstract
In this article, we establish sufficient conditions on the generalized Cesáro and Orlicz sequence spaces mathbb{E} such that the class S_{mathbb{E}} of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers belonging to mathbb{E} generates an operator ideal. The components of S_{mathbb{E}} as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. Some inclusion relations between the operator ideals as well as the inclusion relations for their duals are obtained. Finally, we show that the operator ideal formed by mathbb{E} and approximation numbers is small under certain conditions.
Highlights
The operator ideals theory is gaining importance in functional analysis, since it has many applications in spectral theory, geometry of Banach spaces, eigenvalue distributions theorem, fixed point theorem, etc
Some of operator ideals in the class of Banach spaces or Hilbert spaces are defined by different scalar sequence spaces
Pietsch [1] examined the operator ideals formed by the classical sequence space p (0 < p < ∞) and the approximation numbers
Summary
The operator ideals theory is gaining importance in functional analysis, since it has many applications in spectral theory, geometry of Banach spaces, eigenvalue distributions theorem, fixed point theorem, etc. In [2], the authors studied the operator ideals constructed by generalized Cesáro and Orlicz sequence spaces M and approximation numbers. We give sufficient conditions on Orlicz and generalized Cesáro sequence spaces such that the operator ideal formed by approximation numbers is small.
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