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 ,
and, hence
On the other hand, the operator norm bounds by the essential supremum
of the absolute value . For any , the measure of the
set
is greater than zero. If , set , otherwise let
be a subset of whose measure is finite. Then, if is
the 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 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
and, hence
Since this is true for every , we see that the operator norm is
infinite, i.e. the operator is unbounded.
Title | operator norm of multiplication operator on |
---|---|
Canonical name | OperatorNormOfMultiplicationOperatorOnL2 |
Date of creation | 2013-04-06 22:14:23 |
Last modified on | 2013-04-06 22:14:23 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 13 |
Author | rspuzio (6075) |
Entry type | Theorem |
Classification | msc 47B38 |