Accuracy of approximation of a solution to an abstract Cauchy problem (Q2508779)

Let \(A\) be a closed linear logarithmically-sector operator on a Banach space \(E\), whose domain is dense in \(E\). The authors prove that for each \(x_0 \in D(A^\sigma), \sigma > 1\), the Cauchy problem \[ {dx(t)\over dt} + Ax(t) =0, \;t >0, \quad x(0) = x_0 \] is uniquely solvable, and they provide a series representation for its unique solution in terms of Laguerre polynomials and Cayley transform of the operator \(A\).