Given an arbitrary discretized operator, or matrix A, we consider the matrix element of the resolvent in the complex z-plane g(z) =< uj(A zI) 1ju >, where u is a vector with random components. Since k A zI k>= g(z), isocontours of jg(z)j can be used as lower bounds for the boundaries of the ffl-pseudospectra of A. Unlike the norm of the resolvent, g(z) can be expressed as an analytical function of z and can be subsequently displayed at little cost in any region of the complex plane with arbitrary high resolution. Furthermore, our method uses the Arnoldi algorithm and can thus take advantage of sparse representations of operators.