Let $\cal H$ be a Hilbert space and $T$ be a non-negative invertible self-adjoint (unbounded) operator on $\cal H$. Let $\lambda >0$, and consider $z \in \mathbb{C}$ such that $z=\alpha-i\beta$, with $0\le \beta \le \lambda$. Now consider the operators $T^\lambda,~T^{iz}$ and $T^{\beta}$. Now I want to show that $D(T^\lambda) \subset D(T^\beta)$ and $D(T^\beta)=D(T^{iz})$, where $D(A)$ denotes the domain of definition of the operator $A$ on $\cal H$.
So, since $T$ is non-negative we can define using continuous functional calculus $T^\lambda$ and $T^\beta$ easily as $\lambda,\beta >0$. Now note that $iz=i(\alpha-i\beta)=\beta-i\alpha$. Now I am unable to understand how to define $T^{\beta-i\alpha}$ and also their domain of definitions.
Basically I know that for the unbounded operators domain plays an important role. I started unbounded operator and the aforementioned argument was given in a lecture. I am unable to understand how the operator $T^{iz}$ is defined and how can I understand domains of $T^\lambda,~T^{iz}$ and $T^{\beta}$ and the relation given above between their domains? Please help me to understand this.