# Quasi-compact operators on $\ell^1$

Assume that $$T:\ell^1 \rightarrow \ell^1$$ is a bounded linear operator, given by an infinite matrix $$(a_{ij})_{i,j\in\mathbb{N}}$$ so that $$(Tx)_i = \sum_{j=1}^{\infty} a_{ij}x_j$$. In particular, I am interested in nonnegative matrices ($$a_{ij}\geq0$$) such that $$a_{ii} = 0$$ and $$a_{ij}>0$$ only for $$i-m \leq j \leq i+m$$, $$j \neq i$$, where $$m$$ is a constant.

I would like to understand when such $$T$$ is quasi-compact. Are there sufficient or necessary conditions in terms of the matrix entries? What would be a good intuition? How should a counter-example look like?

• Could you remind us of the definition of quasi-compactness for operators on a Banach space? – Yemon Choi Jul 2 '19 at 22:38
• @YemonChoi: I suspect that "quasi-compact" means that there is a compact operator $K$ on $\ell^1$ and an integer $n \ge 1$ such that $\|T^n-K\| < 1$. – Jochen Glueck Jul 3 '19 at 6:20
• @C.Kawan: Concerning your last question for a counterexample: Are you looking for an operator $T$ of the given form such that $T$ is not quasi-compact? In this case you can just take the left shift operator (or alternatively, the right shift operator) on $\ell^1$. But maybe I misunderstood the question? – Jochen Glueck Jul 3 '19 at 6:24
• In the literature, I can find different definitions of quasi-compactness, and I don't know if they are all equivalent. What I mean by quasi-compactness is sometimes called "spectral gap": The essential spectral radius is strictly less than the spectral radius. I would like to get some intuition for this property and know whether there are concrete criteria how to check it for operators of the form I have described above. – C. Kawan Jul 3 '19 at 9:38
• The case that is the most interesting to me is when the operator $T$ is irreducible, i.e., for every pair $(i,j)$ of indices there is $n$ so that $T^n_{ij} > 0$, where I identify $T$ with its matrix representation. One of simplest cases would be a band matrix $(a_{ij})$ with all entries equal to zero except for $a_{i,i+1}$ and $a_{i,i-1}$ (which should have positive values). Is it possible to give a criterion for quasi-compactness in this case? – C. Kawan Jul 4 '19 at 8:17