Consider the real positive definite block matrix $$ X = \begin{bmatrix} A &B \\ B^T &C \end{bmatrix} $$ with dimensions: $A$ is $d \times d$, $C$ is $k \times k$, and $B$ is $d \times k$, so $X$ is $(d+k) \times (d+k)$. Both $A$ and $C$ are positive definite as well. I am interested in upper bounds on the operator norm $\| X \| = \|X\|_2$ in terms of $A, B, C$. In particular, are there any bounds apart from the general one:
$$ \|X\| \le \|A\| + 2\|B\| + \|C\| $$
that is described here?
As the matrix $X$ is positive definite we have $$\|X\|=\max_{\|v\|_2=1}\langle Xv,v\rangle$$ Let $v=(x,y),$ where $x\in \mathbb{R}^d,\ y\in \mathbb{R}^k.$ Then $$\displaylines{\langle Xv,v\rangle=\langle Ax,x\rangle +\langle Cy,y\rangle+2\langle By,x\rangle \\ \le \|A\|\,\|x\|_2^2+\|C\|\,\|y\|_2^2+2\|B\|\,\|x\|_2\|y\|_2\qquad (*)\\ \le \|A\|\,\|x\|_2^2+\|C\|\,\|y\|_2^2+\|B\|(\|x\|_2^2+\|y\|_2^2)\\ \le \left [\max(\|A\|,\|C\|)+\|B\|\right ]\,\|v\|_2^2} $$ Hence $$\|X\|\le \max(\|A\|,\|C\|)+\|B\|\qquad \qquad\qquad \ (**)$$ The estimate is optimal if $B=0.$
There is still some space for improvement as the quantity $(*)$ can be estimated by the norm of $2\times 2$ matrix of the form $$\begin{pmatrix} \|A\| & \|B\|\\ \|B\| &\|C\|\end{pmatrix} $$ times $\|v\|_2^2.$ In order to calculate the norm, it suffices to find the eigenvalues of this matrix and choose the one with greater absolute value. The eigenvalues of the matrix are equal $${\|A\|+\|C\|\over 2}\pm {1\over 2}\sqrt{(\|A\|-\|C\|)^2+4\|B\|^2}$$ so the norm of the matrix is equal $${1\over 2}[\|A\|+\|C\|]+ {1\over 2}\sqrt{(\|A\|-\|C\|)^2+4\|B\|^2}$$ This estimate is slightly better than $(**).$