Sequences of operators, monotone in the sense of contractive domination
Seppo Hassi, Henk de Snoo
TL;DR
This work develops a comprehensive framework for sequences of monotone operator relations in Hilbert spaces, where contractive domination drives convergence. By employing representing maps and the monotonicity principle, the authors establish existence and structure results for limits of nondecreasing sequences of operators and relations, including preservation of closability and closedness, and detailed descriptions via regular parts and Lebesgue-type decompositions. The paper builds a bridge between operator sequences and nonnegative selfadjoint relations, providing explicit descriptions of square-root domains, limit relations, and the interplay between $T$, $T^{**}$, and $T_{\rm reg}$. The results apply to both operator and relation settings, with illustrative examples and an appendix clarifying the algebraic structure of $T^{*}T$ and $T^{*}T^{**}$ and their regular parts. Overall, the work offers a robust toolkit for analyzing monotone limits in spectral and form-theoretic contexts, with implications for convergence in semibounded forms and operator decompositions.
Abstract
A sequence of operators $T_n$ from a Hilbert space ${\mathfrak H}$ to Hilbert spaces ${\mathfrak K}_n$ which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator $T$ from ${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$. Moreover, the closability or closedness of $T_n$ is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.