Sums of two square-zero selfadjoint or skew-selfadjoint endomorphisms

Abstract: Let

V

be a finite-dimensional vector space over a field

m a t h b b F

, equipped with a symmetric or alternating non-degenerate bilinear form

b

. When the characteristic of

m a t h b b F

is not

2

, we characterize the endomorphisms

u

of

V

that split into

u = a_{1} + a_{2}

for some pair

(a_{1}, a_{2})

of

b

-selfadjoint (respectively,

b

-skew-selfadjoint) endomorphisms of

V

such that

(a_{1})^{2} = (a_{2})^{2} = 0

. In the characteristic

2

case, we obtain a similar classification for the endomorphisms of

V

that split into the sum of two square-zero

b

-alternating endomorphisms of

V

when

b

is alternating (an endomorphism

v

is called

b

-alternating whenever

b (x, v (x)) = 0

for all

x i n V

). Finally, if the field

m a t h b b F

is equipped with a non-identity involution, we characterize the pairs

(h, u)

in which

h

is a Hermitian form on a finite-dimensional space over

m a t h b b F

, and

u

is the sum of two square-zero

h

-selfadjoint endomorphisms.