  We present several operator and norm inequalities for Hilbert space
operators. In particular, we prove that if $A_{1},A_{2},...,A_{n}\in {\mathbb
B}({\mathscr H})$, then
\[|||A_{1}A_{2}^{*}+A_{2}A_{3}^{*}+...+A_{n}A_{1}^{*}|||\leq|||\sum_{i=1}^{n}A_{i}A_{i}^{*}|||,\]
for all unitarily invariant norms. We also show that if
$A_{1},A_{2},A_{3},A_{4}$ are projections in ${\mathbb B}({\mathscr H})$, then
&&|||(\sum_{i=1}^{4}(-1)^{i+1}A_{i})\oplus0\oplus0\oplus0|||&\leq&|||(A_{1}+|A_{3}A_{1}|)\oplus
(A_{2}+|A_{4}A_{2}|)\oplus(A_{3}+|A_{1}A_{3}|)\oplus(A_{4}+|A_{2}A_{4}|)||| for
any unitarily invariant norm.
