Conditions for the existence of a lower bound of the operator norm $\|A^*A\|$ for a linear and continuous operator $A:X \to Y$

Question

Setting: $X$, $Y$ Banach (or Hilbert) spaces, $A: X \to Y$ linear, continuous, injective, $A(X)$ is dense in $Y$ ($A$ is an imbedding operator in a Gelfand triple), $A^*$ is its adjoint

Find: Conditions on $A$ for the existence of a constant $C>0$ such that $\|A^*A\|=\sup_x \frac{\|A^*Ax\|_Y}{\|x\|_X} \geq C$.

• You are free to propose additional assumptions, e.g. compactness of $A$.

• I imagine you could use a spectral theorem to find a lower bound with eigenvalues, afaik it was $\|A^*A\|\geq |\lambda|$ for $\lambda \in \sigma(A^*A)$. Or the optimal lower bound is the spectral radius. I do not really want to assume self-adjointness.

• Of course, unitary operators (i.e. $A^*A=Id$) are a class of operators with this feature.

• I don't know if it helps, but since $A(X)$ is dense in $Y$, we know that $A^*$ and $A^*A$ are injective

Answer 1

If $C$ can depend on $A$ we can take $C=\|A^{*}A\|$. You cannot have $C$ independent of $A$. For example take $K$ to be an injective compact operator with dense range and let $A_n =\frac 1 n K$. Your inequality obviously fails for large $n$.