1:let $ S , T \in B ( H ) $ be positive operators. ( $ B ( H ) $ is bounded operator on $H$).
please help me to prove:
$ r ( S + T ) \geq max \{r( S ) , r ( T) \} $?
($ r ( S ) $ is Spectral radius $ S$.)
2: let $ T \in K (H ) $ be normal. ( $ K( H ) $is compact operator on Hilbert space)
is it right to say:
$T \geq 0$ if only if All its eigenvalues are nonzero؟
thanks.
1) Since $S,T\geq 0$ we have in particular $S,T$ normal and hence $r(S)=\|S\|$ as well as $r(T)=\|T\|$. Further it holds in general in a c*-algebra that for $0\leq x \leq y$ holds $\|x\| \leq \|y\|$, hence $r(S)=\|S\| \leq \|S+T\| = r(S+T)$ and similar for $r(T)$.
2) Do you mean self-adjoint instead of normal. For normal operators this is clearly wrong, of course there do exist operators, which are compact, normal but not self-adjoint and in particular not positive.