Characterizations of Moore-Penrose inverses of closed linear relations in Hilbert spaces

TL;DR

The work extends Moore-Penrose inverse theory to closed linear relations in Hilbert spaces and derives a resolvent criterion for a block operator built from $T^{\dagger}T$, namely $\rho(\mathcal{A}) = \{ \lambda : \lambda^{2} \in \rho(TT^{\dagger}) \cap \rho(T^{\dagger}T) \}$. It establishes foundational identities for closed-range relations, such as $(T^{*}T)^{\dagger} = T^{\dagger}(T^{*})^{\dagger}$ and $(TT^{*})^{\dagger} = (T^{*})^{\dagger}T^{\dagger}$, and shows $(T^{\dagger})^{*} = (T^{*})^{\dagger}$. A key contribution is the exact characterization of the Moore-Penrose inverse of direct sums: $(T_{1} \oplus T_{2})^{\dagger} = T_{1}^{\dagger} \oplus T_{2}^{\dagger}$, with implications like $\gamma(T_{1} \oplus T_{2}) = \min\{\gamma(T_{1}), \gamma(T_{2})\}$. These results generalize the pseudo-inverse theory to multivalued operators and enrich spectral and variational analysis in operator theory.

Abstract

This paper examines the Moore-Penrose inverses of closed linear relations in Hilbert spaces and establishes the result $ρ(\mathcal{A}) = \{λ\in \mathbb{C}: λ^{2} \in ρ(TT^{\dagger}) \cap ρ(T^{\dagger}T)\}$, where $\mathcal{A} = \begin{bmatrix} 0 & T^{\dagger} T & 0 \end{bmatrix}$, with $T$ being a closed and bounded linear relation from a Hilbert space $H$ to a Hilbert space $K$, and $T^{\dagger}$ representing the Moore-Penrose inverse of $T$, the set $ρ(\mathcal{A})$ refers to the resolvent set of $\mathcal{A}$. The paper also explores several interesting results regarding the Moore-Penrose inverses of direct sums of closed relations with closed ranges in Hilbert spaces.

Theorems & Definitions (20)

• Theorem 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Theorem 3.7
• Theorem 3.8
• Theorem 3.9
• Theorem 3.10