(1)はルベーグの収束定理を使うためにうまい優関数を見つける必要があります. やること自体はシンプルなのですが何となく馴染みのない計算で少し苦労しました. (2)は特段詰まるところのない問題です.
Since the function is integrable on for every , we obtain from the dominated convergence theorem .
Hence for every such that , the function for is dominated by the integrable function . Then we can interchange the differentiation and integration of and is differentiable.
Let be a sequence in the unit ball of .
From Arzela-Ascoli theorem, it is enough to show that the sequence is uniformly bounded and equicontinuous in .
From Hölder’s inequality, we have for every ,
where denotes the norm of .
From this, the equicontinuity and uniform boundedness is obvious.
Since the operator is self-adjoint, its eigenvalues must be real.
Take and suppose and . Then it follows that
Since the LHS of (5) is continuous, the function is continuous. Then it follows that LHS of (5) is function and therefore is so.
Differentiating both sides twice in (5), we obtain
Since , it follows that . Then we have
If , from (6), it holds that
for constants . Then from (7), we have and . Hence must be of the form
and we can easily check that this function satisfies (5).
If , it follows from (6) that
for constants . From (7), we have and this contradicts to the assumption .
Therefore the eigenvalues of are .