On unbounded complementable operators
Sachin Manjunath Naik, P. Sam Johnson
TL;DR
This paper extends the notion of complementability to unbounded densely defined operators between Hilbert spaces by formulating a generalized Schur complement $T_{/(M,N)}$ relative to closed subspaces $M,N$ and developing a comprehensive framework built on projections, domain decompositions, and operator factorizations. It establishes foundational results for unbounded $(M,N)$-complementable operators, including necessary and sufficient conditions via range inclusions, domain decompositions $D(T) = T^{-1}(N) + M_2$ and $(T^×)^{-1}(M) + N_2$, along with various equivalent formulations in terms of block operator matrices, projections, and symmetry relations. The work synthesizes elements from Douglas' Lemma and previous bounded-operator theories to enable a robust treatment of spectral and boundary-value problems in infinite dimensions, with potential applications in spectral theory, PDEs and control theory. By introducing the formal adjoint $T^×$ and connecting it to the true adjoint $T^*$ through decomposability conditions, the paper provides a versatile toolkit for analyzing unbounded operators via decomposed domains and ranges. Overall, the results lay groundwork for further exploration of unbounded complementability and its interactions with generalized Schur complements in operator theory and its applications.
Abstract
The concept of complementability is extended from bounded operators to densely defined operators on Hilbert spaces. By introducing appropriate projections and decomposition techniques, a framework is developed for analyzing complementability in this broader context. The results provide new insights into the structure of unbounded operators, contributing to the ongoing development of operator theory.