Hyers-Ulam stability of closed linear relations in Hilbert spaces

TL;DR

The paper extends Hyers-Ulam stability from operators to closed linear relations in Hilbert spaces, establishing that for $T \in CR(H,K)$, Hyers-Ulam stability is equivalent to the closedness of the range $R(T)$ and is intimately linked with the adjoint and the spectral properties of $T^{*}T$, including the equality $\sigma(T^{*}T)\setminus\{0\}=\sigma(T^{*}T|_{(N(T))^{\perp}})\setminus\{0\}$ and $\gamma(T^{*}T)=\gamma(T^{*}T|_{(N(T))^{\perp}})>0$. The work further connects stability to the regular part $T_{op}$, the square root $|T|=(T^{*}T)^{1/2}$, and the Moore-Penrose inverse, showing that $T$ is HUS iff $T^{*}T|_{(N(T))^{\perp}}$ is HUS, and that $T^{\dagger}$ is HUS when $T$ is bounded. In the non-negative self-adjoint setting, these equivalences yield practical criteria for stability via spectral data and reduced minimum modulus. Finally, the paper presents sufficient conditions ensuring stability is preserved under sums and products of HUS-stable linear relations, including matrix- or block-structured constructions, broadening the applicability of Hyers-Ulam stability in operator-theoretic contexts.

Abstract

This paper introduces the concept of Hyers-Ulam stability for linear relations in normed linear spaces and presents several intriguing results that characterize the Hyers-Ulam stability of closed linear relations in Hilbert spaces. Additionally, sufficient conditions are established under which the sum and product of two Hyers-Ulam stable linear relations remain stable.

Theorems & Definitions (20)

• Definition 3.1
• Remark 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Proposition 3.6
• Proposition 3.7
• Theorem 3.8
• Lemma 3.9
• Theorem 3.10