On the closedness of operator pencils

Consider an operator pencil A0+λ1A1+···+λnAn in which, for example (other cases are also considered), A0 is a maximal accretive operator, A1, ..., An are closed accretive operators, and dom A0 ⊂ dom Aj, j = ¯1,n¯. We give a sufficient condition under which it is closed for all λj ≥ 0, j = ¯1,n¯. In case n = 1, domA0 = domA1, and A0, A1 are maximal uniformly accretive, this condition is also necessary. The condition is that the matrix (cos(Ai,Aj))ni,j=0 is uniformly cone positive. Here cos(Ai,Aj) is the cosine of the angle between Ai and Aj. 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 × 2 matrices with operator entries.