# 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 $$T^{*}$$.

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