Abstract
A new class of operators called strong strictly singular operators on normed spaces is introduced. This class includes the class of precompact operators, and is contained in the class of strictly singular operators. Some properties and characterizations for these operators are derived.
Highlights
The spaces X and Y will denote normed spaces, and T : X → Y will denote a bounded linear mapping from a normed space X into a normed space Y in this paper
An operator T is called strictly singular if it does not have a bounded inverse on any infinite dimensional subspace contained in X
It is easy to see as a consequence of the open mapping theorem that a continuous linear transformation T from a Banach space X into a Banach space Y has closed range, if and only if for given x ∈ X, there is an element y ∈ X such that Tx = Ty and ‖y‖ ≤ c‖Ty‖, for some fixed c > 0
Summary
The spaces X and Y will denote normed spaces, and T : X → Y will denote a bounded linear mapping from a normed space X into a normed space Y in this paper. It is easy to see as a consequence of the open mapping theorem that a continuous linear transformation T from a Banach space X into a Banach space Y has closed range, if and only if for given x ∈ X, there is an element y ∈ X such that Tx = Ty and ‖y‖ ≤ c‖Ty‖, for some fixed c > 0 (see [2]). This gives a motivation to define a new class of operators called strong strictly singular operators. (b) T is compact on X if and only if Tis compact on X
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