operator norm of multiplication operator on
The operator norm of the multiplication operator is the essential supremum of the absolute value of . (This may be expressed as .) In particular, if is essentially unbounded, the multiplication operator is unbounded.
For the time being, assume that is essentially bounded.
On the one hand, the operator norm is bounded by the essential supremum of the absolute value because, for any ,
On the other hand, the operator norm bounds by the essential supremum of the absolute value . For any , the measure of the set
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 essentially unbounded. This can be dealt with by a variation on the preceeding argument.
If is unbounded, then for all . Furthermore, for any , we can find such that , where
If , set , otherwise let be a subset of whose measure is finite. Then, if is the characteristic function of , we have
|Title||operator norm of multiplication operator on|
|Date of creation||2013-04-06 22:14:23|
|Last modified on||2013-04-06 22:14:23|
|Last modified by||rspuzio (6075)|