Given $L\in\mathbb{C}^{n\times n}$.
Question 1: On what condition on $L$, there exists unit vector $x\in\mathbb{C}^n$, such that
- $x^*L^*Lx=1$.
Question 2: On what condition on $L$, there exists unit vector $x\in\mathbb{C}^n$, such that
- $\Re(x^∗Lx)<0$ and $\Im(x^∗Lx)=0$, simultaneously.
For question 1, I think the necessary and sufficient condition is that singular values of $L$ are spread across $1$. If singular values of $L$ are all less than $1$, then $\max_{x}x^*L^*Lx=\sigma_{max}(L)<1$. Similarly, if singular values of $L$ are all greater than $1$, then $\min_{x}x^*L^*Lx=\sigma_{min}(L)>1$.
For question 2, lets separate $L$ to hermitian part $L_H=(L+L^*)/2$ and skew-hermitian part $L_S=(L-L^*)/2$, then $L=L_H+L_S$. So $\Re(x^∗Lx)<0$ and $\Im(x^∗Lx)=0$, is equivalent to $x^*L_Hx<0$ and $x^*L_Sx=0$.
I think that $x^*L_Hz<0$ is equivalent to $L_H$ having at least one negative eigenvalue, while $x^*L_Sx=0$ is equivalent to $L_S$ having its eigenvalues spread across $0$. However, I am unable to combine these two observations and find simple condition on $L$ like in question 1.