What does this `sp' notation mean?

Question

I'm reading a paper called "On local spectral radius". Throughout there is this weird notation. In particular they say things like

Lemma 2: Let $N$ be a subset of a Banach space $X$. Then $$ \sup \Big\{ r(K,h) : h \in N \Big\} = \sup \Big\{ r(K,h) : h \in sp(N) \Big\} $$

At first I thought `sp' referred to spectrum, but I'm not familiar with any notions of spectrum of a set. Another usage is $sp\{ A^n x : n \geq 0 \}$.

Any help appreciated.

Answer 1

"sp" is not a notation for the spectrum. "sp" means linear span.

The usual notation for the spectrum is $\sigma$.