Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Solution 1:

On every infinite-dimensional Banach space $X$ the compact operators and the strictly singular operators are closed ideals of $\mathcal{L}(X)$. In the present case these are distinct ideals since every operator $\ell^p \to \ell^q$ for $p \lt q$ is strictly singular by a corollary of Pitt's theorem, but e.g. the inclusion is not compact. So there's an easy answer, but maybe these ideals are considered "trivial".

In section 5 of Commutators on $\ell_\infty$, Dosev and Johnson point out that there's a candidate for a maximal ideal on every infinite dimensional Banach space $X$: Say the identity $I$ factors through $T \colon X \to X$ if there are $A$ and $B$ such that $ATB = I$. Define $$\mathfrak{m} = \{M \in \mathcal{L}(X) \mid \text{the identity does not factor through } T\}$$ then clearly $TM, MT \in \mathfrak{m}$ for all $M \in \mathfrak{m}$ and $T \in \mathcal{L}(X)$. So this is an ideal if it is closed under addition: $\mathfrak{m} + \mathfrak{m} \subseteq \mathfrak{m}$ and in that case it is the maximal ideal and hence it is closed. On the Banach spaces $c_0, \ell_p, L_p$ with $1 \lt p \lt \infty$ it can be shown that $\mathfrak{m}$ actually is an ideal.

Unfortunately, in the case $X = \ell^p \oplus \ell^q$ it turns out that $\mathfrak{m}$ is not closed under addition.

However, there is a variant of this idea that works: Let $\mathfrak{m}_p$ be the set of operators $M$ that can be written as $M = AB$ with $B \colon X \to \ell_p$ and $A\colon \ell_p \to X$. Again, it is clear that $TM,MT \in \mathfrak{m}_p$ whenever $T \in \mathcal{L}(X)$ and $M \in \mathfrak{m}_p$ and it an be shown that for $X = \ell_p \oplus \ell_q$ with $p \neq q$ is closed under addition, so $\mathfrak{m}_p$ an ideal. Now $\mathfrak{m}_p$ is not closed, but its closure $\mathfrak{a}_p$ is an ideal. It is a theorem of H. Porta, Factorable and strictly singular operators, Studia Math. 37 (1971) 237–243, that $\mathfrak{a}_p$ and $\mathfrak{a}_q$ are the only two maximal ideals of $\mathcal{L}(X)$ and that $\mathfrak{a}_p \cap \mathfrak{a}_q$ consists precisely of the strictly singular operators.