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For a negative unbounded operator $T$, what equals the operator $$ T^+ = \left[\frac{1}{2}(|T| + T) \right]^{**},$$ where $|T|= (T^2)^{1/2}$ and $A^{**} $ is the minimal closed extension of an operator $A$.

Thank you in advance

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  • $\begingroup$ what does ** mean? $\endgroup$ Apr 7, 2016 at 15:24
  • $\begingroup$ $A^{**}$ is the minimal closed extension of an operator $A$ $\endgroup$
    – Z. Alfata
    Apr 7, 2016 at 15:33

1 Answer 1

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The adjective "negative" implies that you have an operator on a Hilbert space. It then follows from the spectral theorem for unbounded operators that your expression is simply the zero operator. The double star operation ensures that its domain is the whole space, not just that of $T$.

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  • $\begingroup$ The graph of the algebraic sum is the set $(x,0)$ with $x$ in the domain of $T$. This is not closed, so this sum is not self-adjoint. Its closure is the graph of the zero operator as I stated in my answer. $\endgroup$
    – goleta
    Apr 8, 2016 at 6:44

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