What is the spectrum of the continuous extension of a bounded operator?

Question

If $H$ is a Hilbert space and $H_{0}$ is a dense subspace in $H.$

Giving a bounded operator $A_{0}:H_{0}\rightarrow H_{0}$ with $\sigma \left( A_{0}\right) $ as spectrum, what would be $\sigma \left( A\right) $, if $A$ is the continuous extension of $A_{0}$ to $H$ ?

Thank you !