On unbounded p-summable Fredholm modules

We prove that odd unbounded p-summable Fredholm modules are also bounded p- summable Fredholm modules (this is the odd counterpart of a result of A. Co nnes for the case of even Fredholm modules). The approach we use is via est imates of the form parallel to phi(D) - phi(D-o)parallel to L-p(M,L-tau) le ss than or equal to C . parallel to D - D-o parallel to (1/2), where phi(t) = t(1 + t(2)) (-1/2), D-o = D-o* is an unbounded linear operator affiliate d with a semifinite von Neumann algebra M, D - D-o is a bounded self-adjoin t linear operator from M and (1 + D-o(2)) (-1/2) is an element of L-p(M,tau ), where L-p(M,tau) is a non-commutative L-p-space associated with M. It fo llows from our results that if p is an element of (1, z), then phi(D) - phi (D-o) belongs to the space L-p(M, tau). (C) 2000 Academic Press.