Well-posedness of the cauchy problem for complete second-order operator-differential equations

Abstract

For the equation y″(t)+Ay′(t)+By(t)=0, where A and B are arbitrary commuting normal operators in a Hilbert space H, we obtain a necessary and sufficient condition for well-posedness of the Cauchy problem in the space of initial data D(B)×(D(A)∩D(|B|^1/2)) and for weak well-posedness of the Cauchy problem in H×H_(|A|+|B|^1/2+1). This condition is expressed in terms of location of the joint spectrum of the operators A and B in C ². In terms of location of the spectrum of the operator pencil z ²+Az+B in C ¹, such a condition cannot be written.