Multiplication Operator on Köthe Spaces: Measure of Non-compactness and Closed Range

The essential norm of multiplication operators acting on Orlicz sequence spaces

“…However there are some works about this subject. If X is a non-atomic order continuous Köthe space, Castillo et al [3] proved that the essential norm of M u : X → X is given by kM u k e = kuk ∞ , the essential supremum of |u|. Hence, the only compact multiplication operator acting on this kind spaces is the null operator.…”

“…These authors studied the compactness and closedness of the range of multiplication operators on L 2 (µ). A big numerous of mathematicians have extended or generalized the results of Singh and Kumar to important examples of Köthe spaces (we omit cites), however, the more general study of the properties of the operator M u : X → X (such as the compactness and closedness of the range among others), with X being a Köthe space is due to Hudzik et al [5] (see also Castillo et al [3]).…”

“…This subject has been widely studied in the context of analytic functions. In the case of multiplication operators M u , acting on non-atomic Köthe space, the essential norm was calculate by Castillo et al [3] and it is an open problem calculate the essential norm of this operator in a more general sense. The essential norm of multiplication operators acting on Lorentz sequence spaces (which is a Köthe sequence space) was calculate by Castillo et al [4].…”

On the Essential Norm of Multiplications Operators Acting on Cesàro Sequence Spaces

“…This subject has been widely studied in the context of analytic functions. In the case of multiplication operators acting on nonatomic Köthe spaces the essential norm was calculate by Castillo et al [15] and it is an open problem calculates the essential norm of this operator in a more general sense. The essential norm of multiplication operators acting on Lorentz sequence spaces was calculated by Castillo et al [16].…”

“…This last condition also characterizes the compactness of this operator , acting on other Banach sequence spaces such as Orlicz-Lorentz sequence spaces [10], Cesàro sequence spaces [11], Cesàro-Orlicz sequence spaces [12], among others. However, the above spaces are classified as Köthe sequence spaces and the characterization of compactness (and other properties) of multiplication operators acting on Köthe sequence spaces is due to Ramos-Fernández and Salas-Brown [13] (see also [14,15]). It is an open problem to characterize the multipliers between two Köthe spaces and .…”