I am interested in the set of $n \times n$ complex square matrices $H$ such that $H + H^* = I$. Here, $H^*$ is the conjugate transpose of $H$ and $I$ is the $n \times n$ identity matrix. These matrices are not quite skew-Hermitian, but close. Do they have a name/is there a good reference that expounds their properties?
Incidentally, I know they relate to unitary matrices in the following way, which is how I came across them:
Fact: The difference $U - V$ of two $n \times n$ unitaries $U$ and $V$ is unitary if and only if $U^*V + (U^*V)^* = I$.
I think the matrices are just matrices of the form $U+\frac{1}{2}I$, where $U$ is skew-Hermitian (i.e., $U^*=-U$).
Indeed, such a matrix has $(U+\frac{1}{2}I) + (U+\frac{1}{2}I)^* = U+U^*+I = I$.
And if $H+H^*=I$, then $H-\frac{1}{2}I$ satisfies $$\left(H-\frac{1}{2}I\right) + \left(H-\frac{1}{2}I\right)^* = H+H^* - I = I-I = 0,$$ so $H-\frac{1}{2}I$ is skew-Hermitian, proving that $H=U+\frac{1}{2}I$ for some skew-Hermitian matrix $U$.
The behavior of $A+\mu I$ in terms of the behavior of $A$ is well-understood (same eigenvectors, eigenvalues of $A+\mu I$ are of the form $\lambda+\mu$ where $\lambda$ is an eigenvalue of $A$, etc), so I'm not sure there is anything to say about these matrices beyond what you say about skew-Hermitian matrices.