On the closedness of operator pencils

Consider an operator pencil A(0) + lambda(1)A(1) + ... + lambda(n)A(n) in which, for example (other cases are also considered), A(0) is a maximal accretive operator, A(1), ..., A(n) are closed accretive operators, and dom A(0) subset of dom A(j), j = (1,n) over bar. We give a sufficient condition under which it is closed for all lambda(j) greater than or equal to 0, j = (1,n) over bar. In case n = 1, dom A(0) = dom A(1), and A(0), A(1) are maximal uniformly accretive, this condition is also necessary. The condition is that the matrix (cos(A(i),A(j)))(i,j=0)(n) is uniformly cone positive. Here cos(A(i),A(j)) is the cosine of the angle between A(i) and A(j). We prove some new and reprove some old results related to uniform cone positivity and the cosine. In the final section we study the closedness of some 2 x 2 matrices with operator entries.