Relate $ \langle X\,,\mathcal{A}^{*}\mathcal{A}\,X\rangle $ and $\langle X\,,\mathcal{A}\,X\rangle $

Question

Consider the Frobenius inner products $ \langle X\,,\mathcal{A}^{*}\mathcal{A}\,X\rangle $ and $\langle X\,,\mathcal{A}\,X\rangle $ where $\mathcal{A}$ is some operator and $\mathcal{A}^{*}$ is the adjoint operator.

I am interested in finding a suitable bound for the first inner product. I understand the second inner product fairly well and know a suitable lower bound. Is there something I can say about the sizes of the inner products with respect to one another?

Answer 1

The most obvious estimate that I can think of is \begin{align} \langle X, \mathcal{A}^\ast\mathcal{A} X\rangle =\langle \mathcal{A}X, \mathcal{A} X\rangle = \|\mathcal{A}X\|^2 \leq \|\mathcal{A}\|_{\text{op}}^2\|X\|^2. \end{align}