The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol

Abstract: We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime through an additional parameter, leading to regularized memory effects. We develop a complete symbolic and Volterra theory, including explicit Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. We show that the natural factorization of the Laplace symbol does not fit the classical Bernstein product mechanism and that the symbol is not a Bernstein function in general. Despite this non-Bernstein character, we establish well-posedness of abstract fractional Cauchy problems with sectorial generators by resolvent estimates and Laplace inversion, yielding a W-resolvent family with temporal regularity and smoothing properties. As an illustration, we apply the theory to a W-fractional diffusion model and discuss the influence of the modulation parameter on the relaxation of spectral modes.