Spectral Theorem for quaternionic normal operators: Multiplication form

2016/03/02 by G. Ramesh, Ramesh, G., P. Santhosh Kumar +1 · 1 citation
Mathematics · #35P05 #47B15 #47S10 #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Mathematical Analysis and Transform Methods #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1603.00697

openalex publication_date 2016/03/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let H be a right quaternionic Hilbert space and let T be a quaternionic normal operator with the domain D(T) ⊂ H. Then for a fixed unit imaginary quaternion m, there exists a Hilbert basis N_m of H, a measure space (Ω, μ), a unitary operator U \colon H → L²(Ω; ℍ; μ) and a μ - measurable function ϕ\colon Ω→ ℂ_m (here ℂ_m = \α+ m β; α, β∈ ℝ\) such that Tx = U^*M_ϕUx, for all x∈ D(T), where M_ϕ is the multiplication operator on L²(Ω; ℍ; μ) induced by ϕ with U(D(T)) ⊆ D(M_ϕ). In the process, we prove that every complex Hilbert space is a slice Hilbert space. We establish these results by reducing it to the complex case then lift it to the quaternionic case.

Cited by

• Cowen-Douglas operators on quaternionic Hilbert spaces

Related

• Spectral theorem for unbounded normal operators in quaternionic Hilbert spaces
• Normal operators in real and quaternionic Hilbert spaces
• The Spectral Theorem for Quaternionic Normal Operators
• Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic\n S-Functional Calculus and Spectral Operators
• Perturbation of normal quaternionic operators