A family of operators with everywhere dense graphs (Q5902769)

Families of pathological Hilbert space operators are described. The closure of the graph of each of these operators is the whole of the direct sum Hilbert space. For a locally compact Hausdorff space X on which there is defined a nonatomic regular Borel measure, each family is indexed by the points of X together with the complex-valued functions on the positive integers. This is then used to show: (i) the impossibility of defining operator-valued free quantum fields (ii) the non-differentiability of almost all (sample) paths of Brownian motion.