When I was reading about unbounded operator I found that every m-dissipative operator has closed graph so by closed graph theorem this operator is bounded as we work in Banach space, I am confused because we were studying the unbounded operator, any help, please. Thank you.
2026-03-28 17:00:02.1774717202
Boundedness of $m$-dissipative operator
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The closed graph theorem states if the domain of an operator $A:D(A)\subset X\to X$ is the Banach space $X$, then it is bounded. i.e., $X=D(A)$ implies the boundedness of $A$. That is, an everywhere defined closed operator is bounded. However, in the general case, $D(A)\neq X$. Not all closed operators are everywhere defined. For example, let us take $A:=\Delta$ with domain $D(A):=H^2(\Omega)\cap H_0^1(\Omega)$ is an m-dissipative operator in $L^2(\Omega)$. This operator is not bounded.