Let $V$ be the Volterra operator, $(Vf)(t)=\int_0^t f(s) ds$, acting on the Hilbert space $L_2(0,1)$, and let us denote $A=(I+V)^{-1}$.

Then $\|A\|=1$ and $\sigma(A)=\{1\}$ [Halmos, A Hilbert space problem book, 2nd ed. Problem 190], but $A\neq I$ because $V\neq 0$.

Let $V$ be the Volterra operator, $(Vf)(t)=\int_0^t f(s) ds$, acting on the Hilbert space $L_2(0,1)$, and let us denote $A=(I+V)^{-1}$.

Then $\|A\|=1$ and $\sigma(A)=\{1\}$ [Halmos, A Hilbert space problem book, 2nd ed. Problem 190], but $A\neq I$ because $V\neq 0$.

$A$ has empty point spectrum because so has $V$.