I'm going through the topic C* algebra & operator algebra and facing few questions . It would be great if you people could help me to clear the doubts.
Q4. If T belongs to B(H) such that the resolvent set of T dense in C, the complex number set . does it imply H is finite dimensional ?
Q5. Can we find a non normal T in B(H) such that spectrum set is [-1,1]
No, in any dimension there are nontrivial operators with spectrum $\{0\}$. For instance the Volterra Operator has spectrum $\{0\}$. You can even get a selfadjoint (more: positive) operator, in any dimension, with discrete spectrum; for instance, fix $e\in H$ and define $Te=e$, and zero on $\{e\}^\perp$. Then $\sigma(T)=\{0,1\}$.
And as long as $H$ is infinite-dimensional, you can always find operators with spectrum $[-1,1]$. The easiest way is to represent $H=L^2[-1,1]$ and take $(Tf)(t)=tf(t)$. But this is normal. To get a non-normal one, write $H=H_1\oplus \mathbb C^2$, construct $T_0\in B(H_1)$ normal with spectrum $[-1,1]$, and define $$T=\begin{bmatrix} T_0&0\\0&J\end{bmatrix},\ \ \ \ \text{ with } J=\begin{bmatrix} 0&1\\0&0\end{bmatrix}.$$
