[Paper Review] The reverse order law for Moore-Penrose inverses of operators on Hilbert C*-modules

Kamran Sharifi, Behnaz Ahmadi Bonakdar|arXiv (Cornell University)|Mar 25, 2014

Matrix Theory and Algorithms15 references3 citations

TL;DR

This paper establishes necessary and sufficient conditions for the reverse order law $(TS)^{ ewcommand{}{\dagger}} = S^\f T^\f$ to hold for Moore-Penrose inverses of bounded adjointable operators on Hilbert C*-modules. The key result shows the equality holds if and only if $\operatorname{Ran}(T^*TS) \subseteq \operatorname{Ran}(S)$ and $\operatorname{Ran}(SS^*T^*) \subseteq \operatorname{Ran}(T^*)$, generalizing Greville's matrix result to the noncommutative operator setting.

ABSTRACT

Suppose $T$ and $S$ are bounded adjointable operators between Hilbert C*-modules admitting bounded Moore-Penrose inverse operators. Some necessary and sufficient conditions are given for the reverse order law $(TS)^{ †} =S^{ †} T^{ †}$ to hold. In particular, we show that the equality holds if and only if $Ran(T^{\ast}TS) \subseteq Ran(S)$ and $Ran(SS^{\ast}T^{\ast}) \subseteq Ran(T^{\ast}),$ which was studied first by Greville [{\it SIAM Rev. 8 (1966) 518--521}] for matrices.

Motivation & Objective

• To generalize Greville's reverse order law for matrices to bounded adjointable operators on Hilbert C*-modules.
• To identify necessary and sufficient conditions under which $(TS)^\dagger = S^\dagger T^\dagger$ holds in the noncommutative setting of Hilbert C*-modules.
• To address the lack of orthogonal complementation and standard Hilbert space properties in Hilbert C*-modules by characterizing the Moore-Penrose inverse via range and kernel inclusions.
• To resolve an open problem posed in [17] concerning the reverse order law for modular operators.

Proposed method

• Uses matrix representations of operators relative to orthogonal decompositions of Hilbert C*-modules into kernel and range components.
• Applies the characterization of Moore-Penrose inverses via four defining equations: $TXT=T$, $XTX=X$, $(TX)^*=TX$, $(XT)^*=XT$.
• Employs the equivalence of closed range and existence of bounded Moore-Penrose inverse, as established by Xu and Sheng.
• Translates operator conditions into range and kernel inclusions using the decomposition $T = T_1 \oplus T_2$, $S = S_1 \oplus S_2$ with $T_2, S_2$ representing the kernel parts.
• Derives equivalent conditions via commutator identities and projections, particularly $[T_1T_1^*, D^{-1}] = 0$ and $T_2T_2^*T_1 = 0$, to analyze the structure of $T^\dagger S^\dagger$.
• Establishes equivalence between the reverse order law and two range inclusion conditions: $\operatorname{Ran}(T^*TS) \subseteq \operatorname{Ran}(S)$ and $\operatorname{Ran}(SS^*T^*) \subseteq \operatorname{Ran}(T^*)$.

Experimental results

Research questions

• RQ1 Under what conditions does the reverse order law $(TS)^\dagger = S^\dagger T^\dagger$ hold for bounded adjointable operators on Hilbert C*-modules?
• RQ2 How can Greville’s matrix result on the reverse order law be generalized to the setting of Hilbert C*-modules where standard Hilbert space properties fail?
• RQ3 What role do range and kernel inclusions play in characterizing the Moore-Penrose inverse of a product of operators in noncommutative operator algebras?
• RQ4 Can the reverse order law be equivalently reformulated in terms of projections and commutators in the Hilbert C*-module framework?
• RQ5 How do the properties of the Moore-Penrose inverse in Hilbert C*-modules differ from those in Hilbert spaces, particularly regarding orthogonal decompositions?

Key findings

• The reverse order law $(TS)^\dagger = S^\dagger T^\dagger$ holds if and only if $\operatorname{Ran}(T^*TS) \subseteq \operatorname{Ran}(S)$ and $\operatorname{Ran}(SS^*T^*) \subseteq \operatorname{Ran}(T^*)$.
• The equivalence of the reverse order law to two intermediate conditions—$(TS)^\dagger TS = S^\dagger T^\dagger TS$ and $TSS^\ast T^\dagger T = TSS^\ast$—is established via matrix decomposition and projection analysis.
• The condition $\operatorname{Ran}(T^*TS) \subseteq \operatorname{Ran}(S)$ ensures that the action of $T^*TS$ lies within the range of $S$, which is necessary for the product inverse to factor correctly.
• The condition $\operatorname{Ran}(SS^*T^*) \subseteq \operatorname{Ran}(T^*)$ ensures compatibility between the adjoint of $T$ and the product $SS^*T^*$, preserving the structure of the inverse.
• The proof relies on decomposing operators into range and kernel components and analyzing the resulting matrix blocks, particularly showing $T_2T_2^*T_1 = 0$ is equivalent to the range inclusions.
• The result generalizes Greville’s 1966 matrix result to Hilbert C*-modules, confirming that the same range inclusion conditions are both necessary and sufficient in the noncommutative operator setting.