Let $X=[x_1 \dots x_n] \in \mathbb{R}^{p \times n}$ be the data matrix where the $x_i \in \mathbb{R}^{p \times 1}$. Let the data be "centered at origin", so $\frac{1}{n}\sum_{i=1}^{n}x_i = 0$. Let $C_X=\frac{1}{n}\sum_{i=1}^{n}x_i x_i' \in \mathbb{R}^{p \times p}$ be the sample covariance matrix. Let $Q_X(z):=(C_X + zI_p)^{-1}$ be the resolvent of $C_X, $ and consider it for all $z > 0$. Note that, $Q_X(z)$ is defined for $z > 0,$ as $zI_p + C_X$ is positive definite (hence invertible), because $zI_p$ is positive definite and $C_X$ is positive semidefinite.
I'm trying to prove the following two upper bounds on the operator norm $||*||_{op}$
(1) $||Q_X(z)||_{op} \leq 1/z \forall z > 0$
(2) $||Q_X(z)X||_{op} \leq \sqrt{n} \forall z > 0$
EDIT: To solve (1), I used the fact that the eigenvalues of $Q_X$ are inverse of those of $C_X + zI_p$, and the eigenvalues of $C_X + zI_p$ are of the form $\lambda_i + z,$ where $\lambda_i, 1 \le i \le d$ are eigenvalues of $C_X$. Then we have: $||Q_X(z)||_{op}=$ maximum eigenvalue of $Q_X(z) = max_{1 \leq i \leq s} \frac{1}{\lambda_i + z} \leq 1/z.$
What about for the second problem?
Any hints or solutions appreciated!