Let $T\neq\text{id}$ be a positive invertible operator on an infinite dimensional Hilbert space. Can $\|T\|$ belong to the point spectrum of $T$?

Let $G$ be a l.c. group and $f$ belong to $C_c(G)$, the space of continuous functions with compact support. Define an operator$T_f$ on $L^2(G)$ by $T_f(g)=f*g$ (the convolution product). If $T_f$ is positive and invertible, could $\|T_f\|$ belong to the point spectrum of $T_f$?

Let $T$ be a positive invertible operator on an infinite dimensional Hilbert space. Can $\|T\|$ belong to the point spectrum of $T$?

Let $T\neq\text{id}$ be a positive invertible operator on an infinite dimensional Hilbert space. Can $\|T\|$ belong to the point spectrum of $T$?