We introduce the Tracy-Singh sum for operators on a Hilbert space, generalizing both the Tracy-Singh sum for
matrices and the tensor sum for operators. The Tracy-Singh sum is shown to be compatible with algebraic operations and order
relations. Then we establish a binomial theorem involving Tracy-Singh sums, and its consequences. We also investigate
continuity, convergence, and norm bounds for Tracy-Singh sums. Moreover, we derive operator identities involving Tracy-Singh
sums and certain operator functions defined by power series