Evolution equations associated with positive linear operators

We study a general class of Szász–Mirakjan–Durrmeyer type operators $S_{n, j}$ acting on exponentially bounded functions on $[0, \infty)$, with emphasis on their interpretation through an associated degenerate parabolic Cauchy problem. Using the evolution viewpoint together with differentiation properties of the operator family, we derive evolution equations satisfied by spatial derivatives of the generated solution and obtain propagation results for sign conditions on derivatives, leading to higher order shape-preserving properties. We also discuss explicit eigenstructures and separated-variable solutions, highlighting the role of classical special functions. The action of $S_{n, j}$ on exponential functions is employed to compute moments via differentiation with respect to the exponential parameter, yielding concrete formulas for low-order moments. A main focus is the interaction with $\{e_0, e_j\}$-convexity: we establish pointwise domination and monotonicity-in-time principles for $\{e_0, e_j\}$-convex data, compare $S_{n, j}$ with a discrete modification preserving the same test functions, and derive a Hermite–Hadamard type inequality adapted to $\{e_0, e_j\}$-convex functions, recovering the classical inequality as a particular case.