The decomposable numerical radius and numerical radius of a compound matrix

AbstractLet A be an n × n complex matrix with singular values α1 ⩾ α2 ⩾ ⋯ ⩾ αn and eigenvalues λ1, λ2,…,λn, where |λ1| ⩾ |λ2| ⩾ ⋯ ⩾ |λn|. Denote by Cm (A) (1 ⩽ m ⩽ n) the mth compound of A, and by ⋀mCn the mth Grassmann space over Cn, in which the elements are regarded as complex row vectors with nm coordinates. We have the relation ∏i=1m|λi| ⩽ rd (Cm(A)) ⩽ r(Cm(A)) ⩽ ∏i=1m αi where rd (Cm(A)) = max|xCm(A)x∗| : x is decomposable in ΠmCn, xx∗ = 1 and rd (Cm(A)) = max|xCm(A)x∗| : x ϵ ΠmCn, xx∗ = 1 are the decomposable numerical radius and numerical radius of Cm (A) respectively. In this note we classify those matrices for which Πmi=1|λi| = rd(Cm(A)), Πm i=1|λi| = r(Cm(A)), rd(Cm(A)) = r(Cm(A)), etc. Some of these results answer the questions raised by Marcus and Andresen concerning the decomposable numerical radius of a complex matrix