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
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
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 $T$.