Is either $A$ or $B$ a compact operator?

Question

$X$ is a Banach space, consider two operator $A,B\in B(X)$. Now product of two operator, i.e. $AB$ is compact.

Can we say that one of $A$ and $B$ is compact?

I am just thinking in the following way;

$AB=\frac{1}{4} [(A+B)^2 -(A-B)^2]$

Then, $A$ and $B$ have to be compact.

Answer 1

Although there are exceptions, for most Banach spaces $X$ one can find noncompact operators $A,B:X\to X$ such that $AB=0$. For instance, suppose $X$ is decomposable, that is $X = V\oplus W$ for two infinite dimensional closed subspaces. Let $A:X\to V$ be a projection onto $V$ and $B = I-A$. Neither $A$ nor $B$ is compact, but $AB=0$ is compact. Clearly, any Hilbert space is decomposable.

One of the exceptions is the Argyros-Haydon space. Every bounded linear operator on this Banach space is of the form $\lambda I + K$, where $\lambda\in\mathbb{C}$ and $K$ is a compact operator. For this space, if $AB$ is compact, then either $A$ or $B$ is compact.