1
$\begingroup$

Is there a Banach space $X$, $S$ an unbounded operator defined on a dense subspace $D$ of $X$ and a bounded operator $T$ on $X$ such that

$S+T|_D$

is bounded? What if $T$ is assumed to be compact?

  • 0
    General fact: If $G$ is a group, $H \le G$ is a subgroup, $h \in H$, and $k \notin H$, then $kh, hk \notin H$. The set of all linear operators on $D$ is an (abelian) group under addition, and the bounded operators are a subgroup. So are the compact operators. This is also the fact that an even number plus an odd number is odd.2012-03-12

1 Answers 1

1

Hint: the difference of two bounded operators is bounded.