Definition and analyticity of $T^z$ where $T$ is a positive operator

Question

Let $H$ be a Hilbert space.

Suppose that $T\colon D(T) \to H$ is a positive selfadjoint operator where $D(T)$ is the domain of $T$. The spectrum $\sigma(T)$ of the operator $T$ is a subset of $[0,\infty)$.

1) Sometimes, I see in some papers the operator $T^z$ where $z \in \mathbb{C}$. In the case where $0 \not \in \sigma(T)$, I understand the definition using the functional calculus. My problem is the case where $0 \in \sigma(T)$.

What is the definition of $T^z$, where $z \in \mathbb{C}$, in the case where $0 \in \sigma(T)$?

2)

Under what conditions the map $z \mapsto T^z$ is analytic?

Answer 1

Since $T$ is self-adjoint, you can use the spectral theorem (see e.g. Theorem 13.24 in Rudin, "Functional Analysis").

If $\text{Re}(z) > 0$, $f(\lambda) = \lambda^z$ is a continuous function on $[0,\infty)$.
If $\text{Re}(z) \le 0$, $f(\lambda)$ is not defined at $\lambda=0$, but this is only a problem if $0$ is an eigenvalue rather than just a point in the spectrum.