Question about resolvent set and spectrum

Question

I'm trying to find the solution of this one:

Let $\mathcal{H}$ a Hilbert space and $T \in \mathcal{B}(\mathcal{H})$. Show that

1) $\lambda \in \rho (T) \Longleftrightarrow \bar{\lambda} \in \rho (T^*)$

2) $\sigma (T^*) = \{\lambda \in \mathbb{C} : \bar{\lambda} \in \sigma (T)\}$

3) $ \lambda \in \rho (T) \Longrightarrow R_{T^*} (\bar{\lambda}) = R_T(\lambda)^*$.

I tried to write down the definition of resolvent set and spectrum as follows:

\begin{equation} \rho(T) = \{\lambda \in \mathbb{C}: \exists(T-\lambda I)^{-1} \in \mathcal{B}(\mathcal{H})\} \end{equation} and \begin{equation} \sigma(T) = \mathbb{C}-\rho(T) \end{equation}

But I can't figure out what to do..

Answer 1

All what we need is: if $A \in \mathcal{B}(\mathcal{H})$, then

$A$ is bijective $ \iff A^*$ is bijective

and if $A=T - \lambda I$, then $A^*=T^*- \overline{\lambda} I.$

Can you proceed ?