Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$.

I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two nontrivial closed two-sided ideals.

share|improve this question
add comment

1 Answer 1

up vote 5 down vote accepted

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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.