Strictly singular and strictly co-singular inclusions between symmetric sequence spaces

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Strict singularity and strict co-singularity of inclusions between symmetric sequence spaces are studied. Suitable conditions are provided involving the associated fundamental functions. The special case of Lorentz and Marcinkiewicz spaces is characterized. It is also proved that if E↪ F are symmetric sequence spaces with E≠ℓ 1 and F≠ c 0 and ℓ ∞ then there exist a intermediate symmetric sequence space G such that E↪ G↪ F and both inclusions are not strictly singular. As a consequence new characterizations of the spaces c 0 and ℓ 1 inside the class of all symmetric sequence spaces are given.