(Solved): The adjoint of a linear operator A:x->Y, denoted A^(*), satisfies (:A(x),y:)=(:x,A^(*)(y):) for a ...

The adjoint of a linear operator A:x->Y, denoted A^(*), satisfies (:A(x),y:)=(:x,A^(*)(y):) for any xinx and yinY (note that the inner products on the left and righthand side of the equality above may not be the same operation). (a) Show that if V=A@A^(*), then N(V)=N(A^(*)). (b) Determine the adjoint of A(u)=\int_(t_(0))^(t_(f)) \phi (t_(f),\tau )B(\tau )u(\tau )d\tau (c) Compute the composition V=A@A^(*):R^(n)->R^(n) for the operator in (b). (d) How are the controllability Grammian and the operator V related? Comment in this result in relation to Theorem 5.2 of the Basar text.