operator norm of multiplication operator on
The operator norm of the multiplication operator is theessential supremum of the absolute value
of . (This may beexpressed as .)In particular, if is essentially unbounded
, the multiplicationoperator is unbounded.
For the time being, assume that is essentially bounded.
On the one hand, the operator norm is bounded by the essentialsupremum of the absolute value because, for any ,
and, hence
On the other hand, the operator norm bounds by the essential supremumof the absolute value . For any , the measure of theset
is greater than zero. If , set , otherwise let be a subset of whose measure is finite. Then, if isthe characteristic function of , we have
and, hence
Since this is true for every , we must have
Combining with the inequality in the opposite direction,
It remains to consider the case where is essentiallyunbounded. This can be dealt with by a variation on the preceedingargument.
If is unbounded, then for all . Furthermore, for any , we can find suchthat , where
If , set , otherwise let be a subset of whose measure is finite. Then, if isthe characteristic function of , we have
and, hence
Since this is true for every , we see that the operator norm isinfinite, i.e. the operator is unbounded.