Some conditions are introduced which imply pointwise convergence theorems for increasing sequences of orthogonal projections on $L^2(\mu), \mu$ finite, as well as a pointwise ergodic theorem for self-adjoint and unitary contractions. These results generalize to the case of nonpositive operators some theorems of E. M. Stein.
"Pointwise Convergence Theorems for Self-Adjoint and Unitary Contractions." Ann. Probab. 5 (4) 622 - 626, August, 1977. https://doi.org/10.1214/aop/1176995773