A bounded linear operator $T$ on a Hilbert space (or a Banach space) is said to be normaloid if the operator norm $\| T \|$ of $T$ equals the spectral radius $r(T)=\sup\{ |z| \ | \ z \in \sigma (T) \}$ of $T$ where $\sigma (T)$ denotes the spectrum of $T$. $T$ is said to be hereditarily normaloid if the restriction $T|_{\M}$ of $T$ to every of its invariant subspaces $\M$ is normaloid.
Also, $T$ is said to be totally hereditarily normaloid if $T$ is hereditarily normaloid and for every invertible restriction
$T|_{\M}$ of $T$ to its invariant subspace $\M$ the inverse $(T|_{\M})^{-1}$ is also normaloid.
We shall show that every hereditarily normaloid has the single valued extension property (SVEP)
and hence satisfies Browder's theorem. Also, we give an example of hereditarily normaloid
which is not totally hereditarily normaloid (hence, not paranormal) and does not satisfy Weyl's theorem.