**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 in the original question** 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?)

**Modified item 2** is false; see Willie Wong’s comment on this answer.

**Item 3** is false.  A simple counterexample is <i>n</i>=2, <i>i</i>=1,
$A=\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}$,
$B=\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}$,
$C=\begin{pmatrix}0 & 0 \\ 0 & 1\end{pmatrix}$.
Then $2\sqrt{\lambda_1(BB^*)}=2$ but <i>λ</i><sub>1</sub>(<i>A</i>+<i>C</i>)=1.

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.