On the closedness of operator pencils

Consider an operator pencil A(0) + lambda(1)A(1) + ... + lambda(n)A(n) in w hich, for example (other cases are also considered), A(0) is a maximal accr etive 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 un der 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 maxima l uniformly accretive, this condition is also necessary. The condition is t hat 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 s ome 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 mat rices with operator entries.