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I have come across two different definitions for closable linear operators.

  1. A densely defined linear operator $A$ is called closable if its adjoint $A^\ast$ is also densely defined.
  2. Suppose that there exists a closed linear operator $B$ that extends a linear operator $A:D(A) \to H$. Then $A$ is closable.

Can an 'if and only if' relationship be shown between 1. and 2. or there some differences in these definitions?

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    $\begingroup$ Yes, these are equivalent (for densely defined operators, (2) would work more generally of course). $\endgroup$ – user138530 Feb 10 '17 at 4:23

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