2023/12/12 by Xiaofeng Zhang, Xiaoyi Tian, Zhang, Xiaofeng +3
Computer Science · Mathematics · #46L08 #47A05 #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2312.07257
openalex publication_date 2023/12/12 · openalex created_date 2023/12/14 · openalex updated_date 2026/07/28
This paper deals mainly with some aspects of the adjointable operators on Hilbert C^*-modules. A new tool called the generalized polar decomposition for each adjointable operator is introduced and clarified. As an application, the general theory of the weakly complementable operators is set up in the framework of Hilbert C^*-modules. It is proved that there exists an operator equation which has a unique solution, whereas this unique solution fails to be the reduced solution. Some investigations are also carried out in the Hilbert space case. It is proved that there exist a closed subspace M of certain Hilbert space K and an operator T∈ \mathbbB(K) such that T is (M,M)-weakly complementable, whereas T fails to be (M,M)-complementable. The solvability of the equation A:B=X^*AX+(I-X)^*B(I-X) (X∈\mathbbB(H)) is also dealt with in the Hilbert space case, where A,B∈ \mathbbB(H) are two general positive operators, and A:B denotes their parallel sum. Among other things, it is shown that there exist certain positive operators A and B on the Hilbert space ℓ2(ℕ)⊕ ℓ2(ℕ) such that the above equation has no solution.