4 edited body

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 ||⋅|| (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 A=C=I/2, B=0, and k=1, where 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 n=2, 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$2\sqrt{\lambda_1(B^*B)}=2$ but λ1(A+C)=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. 3 added answers to modified items 2 and 3 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 ||⋅|| (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 A=C=I/2, B=0, and k=1, where 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 n=2, 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 λ1(A+C)=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. 2 clarified unclear notation; made the reference more specific Item 1 is true. This is part of Problem 22 on page 217 (b) in Section 3.5 of Horn and Johnson [HJ94] HJ94], which states that for Ky Fan norm ||⋅|| (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 $\left\|\begin{pmatrix}A & B \\ B^* & C\end{pmatrix}\right\| \le \|A\|+\|C\|$. Instantiate this with the Ky Fan norm to get the desired inequalityleft\|\begin{pmatrix}A & 0 \\ 0 & 0\end{pmatrix}\right\| + \left\|\begin{pmatrix}0 & 0 \\ 0 & C\end{pmatrix}\right\|$.

(I do not have the access to the book right now, and I used the paper by Audenaert [Aud06] to get this reference. [Aud06] [Aud06] contains a proof of a slight generalization of this inequality.inequality among other results.)

Item 2 is false by considering the case where A=C=I/2, B=0, and k=1, where 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.

1