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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?

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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 $V$.

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