Let $A=A_1oplus A_2$. Show that
(1). $W(A)$ is the convex hull of $W(A_1)$ and $W(A_2)$; i.e., the smallest convex set containing $W(A_1)cup W(A_2)$.
(2). $$eex ea sen{A}&=maxsed{sen{A_1},sen{A_2}},\ spr(A)&=maxsed{spr(A_1),spr(A_2)},\ w(A)&=maxsed{w(A_1),w(A_2)}. eea eeex$$
Solution.
(1). We have $$eex ea W(A)&=sed{x^*Ax;sen{x}=1}\ &=sed{y^*A_1y+z^*A_2z;sen{y}^2+sen{z}^2=1}\ &supset W(A_1)cup W(A_2), eea eeex$$ and $$ex W(A)=sed{sen{y}^2 sex{frac{y}{sen{y}}}^*A_1frac{y}{sen{y}} +sen{z}^2 sex{frac{z}{sen{z}}}^*A_2frac{z}{sen{z}}; sen{y}^2+sen{z}^2=1} eex$$ is contained in any convex set containing $W(A_1)cup W(A_2)$.
(2). $$eex ea sen{Ax}^2&=sen{sex{A_1yatop A_2z}}^2quadsex{x=sex{yatop z}}\ &=sen{A_1y}^2+sen{A_2z}^2\ &leq sen{A_1}^2sen{y}^2+sen{A_2}^2sen{z}^2\ &leq maxsed{sen{A_1},sen{A_2}}^2 sex{sen{y}^2+sen{z}^2}\ &=maxsed{sen{A_1},sen{A_2}}^2 sen{x}^2. eea eeex$$ $$eex ea &quad Ax=lm xquadsex{x eq 0}\ & a A_1y=lm y,quad A_2z=lm zquadsex{x=sex{yatop z}}\ & a |lm|leqsedd{a{ll} spr(A_1),&y eq 0\ spr(A_2),&z eq 0 ea}\ & a |lm|leq maxsed{spr(A_1),spr(A_2)};\ &quad A_1y=lm yquadsex{y eq 0}\ & a Asex{yatop 0}=lm sex{yatop 0}\ & a |lm|leq spr(A);\ &quad A_2z=lm zquadsex{z eq 0}\ & a |lm|leq spr(A). eea eeex$$ $$eex ea w(A)&=sup_{sen{x}=1}sev{sef{x,Ax}}\ &=sup_{sen{y}^2+sen{z}^2=1} sev{sef{y,A_1y}+sev{z,A_2z}}\ &leq sup_{sen{y}^2+sen{z}^2=1} sez{ sen{y}^2w(A_1)+sen{z}^2w(A_2) }\ &leq maxsed{w(A_1),w(A_2)};\ w(A_1)&=sup_{sen{y}=1}sen{sef{y,A_1y}}\ &=sup_{sen{sex{yatop 0}}=1} sev{sef{sex{yatop 0},Asex{yatop 0}}}\ &leq w(A),\ w(A_2)&leq w(A). eea eeex$$