Convergence of Complementable Operators
Sachin Manjunath Naik, P. Sam Johnson
TL;DR
This work extends the Schur complement to infinite-dimensional Hilbert spaces through complementable operators and analyzes the convergence behavior of sequences and series of such operators. It introduces the framework of $(M,N)$-complementability, the bilateral Schur complement $T_{/(M,N)}$, and the sets $\\psi(M,N,\\lambda)$ to study when limits preserve complementability, providing both counterexamples (norm-nonclosedness) and sufficient conditions (e.g., $\\lambda_n \\|D-D_n\\| \\to 0$) for stability under limits. The paper also examines the convergence of powers and series $\\sum_n \\alpha_n T^n$, establishing radius-type conditions under which complementability is preserved, and analyzes the topological structure and boundary properties of the set of complementable operators under the strong operator topology. Collectively, these results illuminate when complementability is stable under perturbation, and map the topological landscape governing complementary decompositions in infinite dimensions.
Abstract
Complementable operators extend classical matrix decompositions, such as the Schur complement, to the setting of infinite-dimensional Hilbert spaces, thereby broadening their applicability in various mathematical and physical contexts. This paper focuses on the convergence properties of complementable operators, investigating when the limit of sequence of complementable operators remains complementable. We also explore the convergence of sequences and series of powers of complementable operators, providing new insights into their convergence behavior. Additionally, we examine the conditions under which the set of complementable operators is the subset of set of boundary points of the set of non-complementable operators with respect to the strong operator topology. The paper further explores the topological structure of the subset of complementable operators, offering a characterization of its closed subsets.