functional analysis – Let the complex hilbert space be $H=mathcal{L}^{2}[0,frac{pi}{2}]$. Consider application T and prove the following.

begin{equation}
forall tin left(0,frac{pi}{2} right) qquad qquad Tf(t)=cos (t)int_{0}^{t}sin(s)f(s)ds
end{equation}

  1. Show that $ Tf $ is a continuous function with $ f in H $. Calculate the $ Tf (t) $ values ​​for $ t = 0 $ and $ t = frac{pi}{2} $.

  2. Show that $ T $ is continuous.

  3. Determine $T^*$. Show that $ T $ and $ T^* $ are compact.

  4. Show that if $ f $ is continuous then $ Tf $ and $ T^*f $ are of class $ C^1 $. Calculate the derivatives of $Tf$ and $T^*f$.

  5. Let’s consider $ A = T + T^* $. Show that $ A $ is compact and self-adjoint.

  6. Show that if $ f in H $ and $ Af = lambda f $, with $ lambda $ not null, then $ f $ is of class $ C ^ {2} $ over $ (0, frac {pi } {2}) $ and verify a second-order linear differential equation of the form $ f ” + alpha (lambda) f = 0 $, with $ alpha (lambda) in mathbb {R} $ . Determine $ f (0) $ and $ f (frac{pi} {2})$.

  7. Deduct the eigenvalues ​​of $ A $. Calculate the norm of $ A $.

I made the literal 1), it assumed the existence of a convergent sequence $ (t_ {n}) in (0, frac { pi} {2}) $ and with thanks to the continuity of the cos and the integral it is It follows that $ lim_{n rightarrow infty} Tf (x_ {n}) = Tf (x) $ so Tf is continuous. And evaluate Tf (t) at each point I got zero.

For the literal 2), I used the fact that T is linear, and to show that it is continuous, it sufficed to show that it is bounded like this
begin{equation}
|Tf(t) | = | cos (t) int_{0}^{t}sin(s)f(s)ds | leq |cos (t)| int_{0}^{t} | sin(s) | |f(s)|dsleq int^{t}_{0}|f(s)|dsleq |1|_{L^{2}}|f|_{L^{2}}=sqrt{frac{pi}{2}}||f||_{L^{2}}
end{equation}

por tanto $T$ es continuo.

Now I have a problem from literal 3) to find $ T^{*} $.