**Item 1** is true.  This is part of Problem 22 (b) in Section 3.5 of Horn and Johnson [HJ94], which states that for [Ky Fan norm](http://en.wikipedia.org/wiki/Singular_value_decomposition#Ky_Fan_norms) ||⋅|| (and in fact for any unitarily invariant norm) and a positive semidefinite block matrix $\begin{pmatrix}A & B \\ B^* & C\end{pmatrix}$, it holds that $\left\|\begin{pmatrix}A & B \\ B^* & C\end{pmatrix}\right\| \le \left\|\begin{pmatrix}A & 0 \\ 0 & 0\end{pmatrix}\right\| + \left\|\begin{pmatrix}0 & 0 \\ 0 & C\end{pmatrix}\right\|$.

([Aud06] contains a proof of a slight generalization of this inequality among other results.)

**Item 2** is false by considering the case where <i>A</i>=<i>C</i>=<i>I</i>/2, <i>B</i>=0, and <i>k</i>=1, where <i>I</i> is the identity matrix.  (Did you mean to square the left-hand side?)

References

[Aud06] Koenraad M. R. Audenaert.  A norm compression inequality for block partitioned positive semidefinite matrices.  _Linear Algebra and its Applications_, 413(1):155–176, Feb. 2006.  http://dx.doi.org/10.1016/j.laa.2005.08.017

[HJ94] Roger A. Horn, Charles R. Johnson.  _Topics in Matrix Analysis_.  Cambridge University Press, 1994.