I am curious to know the answer to the following question:


Does there exist a continuous linear operator on some Banach space $X$ such that $\Vert T \Vert=1$, and $\sigma(T)\supset \{1\}$ is isolated in the spectrum of $T$ even though $\{1\}$ is not in the point spectrum? Or does an operator like that not exist?



Existence of operator with certain properties
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 $V$. 


