Factorization, majorization, and domination for linear relations

2014/11/21 by Seppo Hassi, Hassi, Seppo, Henk de Snoo +1 · 2 citations
Computer Science · Engineering · #47A06 #47A63 #47B25 #47B65 #Advanced Graph Theory Research #FOS: Mathematics #Functional Analysis (math.FA) #Matrix Theory and Algorithms #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1411.5922

openalex publication_date 2014/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \mathfrak H_A, \mathfrak H_B, and \mathfrak H be Hilbert spaces. Let A be a linear relation from \mathfrak H to \mathfrak H_A and let B be a linear relation from \mathfrak H to \mathfrak H_B. If there exists an operator Z ∈ B(\mathfrak H_B,\mathfrak H_A) such that ZB ⊂ A, then B is said to dominate A. This notion plays a major role in the theory of Lebesgue type decompositions of linear relations and operators. There is a strong connection to the majorization and factorization in the well-known lemma of Douglas, when put in the context of linear relations. In this note some aspects of the lemma of Douglas are discussed in the context of linear relations and the connections with the notion of domination will be treated.

Cited by

• Product of nonnegative selfadjoint operators in unbounded settings
• An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators

Related

• Mixing Douglas' and weak majorization and factorization theorems
• Factorizations of Linear Relations
• Factorizations of linear relations by idempotents
• Hyers-Ulam stability of closed linear relations in Hilbert spaces
• Lebesgue type decompositions for linear relations and Ando's uniqueness criterion