Does there exists $k\in(0,\infty)$ s.t. $A^*A-I-kH_A>0$?

Question

Given fixed $A\in\mathbb{C}^{n\times n}$, I want to check whether there exists some $k\in(0,\infty)$ such that

$$A^*A-I-kH_A>0,$$

where $H_A=(A+A^*)/2$ and $I$ is an identity matrix.

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Let $A_1=A^*A-I$ and $A_2=H_A$, then we have $A_1-kA_2>0$. Then I think we need to check whether eigenvalues of $A_1-kA_2$ are larger than $0$ for some $k\in(0,\infty)$.

Answer 1

Let $a >0$. If $A=aI$ then your inequality says $a^{2}I-I-kaI >0$ which does not hold for any $k >0$ if $a<1$.