# Ideal of strictly singular operators

Let $$X$$ be a Banach space. An operator $$T:X\to X$$ is called strictly singular iff for any infinite dimensional subspace $$Y\subseteq X,$$ $$T|_{Y}:Y\to T(Y)$$ is not an isomorphism.

It is known that for $$X=\ell_p,$$ $$1\leq p<\infty,$$ an operator is strictly singular iff it is compact. Also $$T:\ell_\infty\to\ell_\infty$$ is strictly singular iff $$T$$ is weakly compact. Can someone provide me proofs for these facts? I could not really locate proofs of the above mentioned facts in literature.

[J. Lindenstrauss and L. Tzafriri. Classical Banach spaces I. Sequence spaces. Springer 1977]. In page 76, after Prop. 2.c.3, it says that the proof of 2.c.3 shows that an operator $$T:\ell_p\to\ell_p$$ is strictly singular if and only if it is compact.
[F. Albiac and N. Kalton. Topics in Banach space theory. Springer 2006] Theorem 5.5.1 says that a weakly compact operator $$T:C(K)\to X$$ is strictly singular, and Theorem 5.5.3 says that a non-weakly compact operator $$T:C(K)\to X$$ is not strictly singular.
Note that $$\ell_\infty$$ is a $$C(K)$$ space with $$K$$ the Stone-Cech compactification of the set of positive integers.