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

TL;DR

This paper introduces the W-operator, a novel two-parameter fractional time operator with a Volterra structure, which modifies low-frequency behavior while maintaining high-frequency properties, and develops its theoretical framework and applications.

Contribution

It presents a new fractional operator with a generalized Laplace symbol, including its symbolic, Volterra, and calculus theories, and demonstrates well-posedness of related fractional Cauchy problems.

Findings

The W-operator preserves high-frequency behavior and allows controlled low-frequency modification.

The operator's symbol is not a Bernstein function, indicating novel spectral properties.

Application to a fractional diffusion model shows the impact of parameters on spectral relaxation.

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.