# Math Help - spectrum of adjoint operator

1. ## spectrum of adjoint operator

Show that $\sigma(T^{*})=\{ \overline{\lambda}: \lambda\in \sigma(T)\}$

where $\sigma(T)$ is the spectrum of $T\in B(H)$ and $B(H)$ is the space consisting of all linear operators mapping elements fromthe Hilbert space $H$ to itself

here's an idea

$\lambda \in \rho(T^{*})<=>(\lambda\mathbb{I} - T^{*})$ exists, and $\rho(T^{*})$ is the resolvant of $T^{*}$
$<=>(\lambda - T^{*})x\neq 0 \ ,\forall x\in H$
$<=>\lambda x \neq T^{*}x, \forall x\in H$
$<=>\overline{\lambda}x\neq x^{*}T, \forall x^{*}\in H$

is this on the right track?

2. I would suggest you think of it this way: if $\lambda \not \in \sigma (T)$, i.e., if $\lambda \in \rho (T)$, then there exists $S\in B(H)$ such that

$T_\lambda S=ST_\lambda=I,$

where $T_\lambda \equiv T-\lambda I$ (or ( $\lambda I-T)$, doesn't matter). That's just the definition of resolvent. Can you see how this translates to in terms of $(T_\lambda S)^*$ and $(ST_\lambda)^*$?