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Showing the null space of a self-adjoint operator is 1-dimensional and the eigenvalues are $\lambda
eq 0$ (Kato’s Inequality)

I’ve got this question that I’ve been working on for a while now but I don’t have a clue on how to approach it.

If $T$ is a self-adjoint linear operator on a complex Hilbert space $H$ with the property that $\ker T = \{0\}$ and $0
ot\in \sigma(T)$, prove that $0
ot\in \sigma(T)$ and $\ker T = \mathbb{C}$.

I do not know how to approach this question. Can anyone help me with how to start this problem?

Suppose that $\ker T
eq \{0\}$. Then there exists $u$ in the kernel, $\|u\| = 1$. If $Tu = 0$, then since $T$ is self-adjoint, $T^*u = 0$.
Consider the case of the complex Hilbert space $H$ having a basis $\{u,v\}$ in which $Tu = 0$, $Tv = 1$, and $v
eq 0$. Since $v
eq 0$, $v = \alpha_1u + \alpha_2v$, where $\alpha_1,\alpha_2 \in \mathbb{C}$. We then have $Tv = \alpha_1Tu + \alpha_2$, but from the equation $T^* = T$, we have
$$
0 = \langle Tv, v \rangle = \alpha_1^* \langle T u, u \rangle + \alpha_2^* \langle T v, v \rangle
$$
and $\langle Tu, u \rangle = 0$, so $\alpha_2 = 0$ and therefore $\alpha_1 = 0$. This is absurd, since $Tv = \alpha_1u + \alpha_2v = \
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