Spectral theorem for unbounded normal operators in quaternionic Hilbert spaces

2015/09/10 by Ramesh, G., Kumar, P. Santhosh
#47B15 35P05 #47S10 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1509.03007

Abstract

In this article, we prove the following spectral theorem for right linear normal operators (need not to be bounded) in quaternionic Hilbert spaces: Let T be an unbounded right quaternionic linear normal operator in a quaternionic Hilbert space H with domain D(T), a right linear subspace of H and fix a unit imaginary quaternion, say m. Then there exists a Hilbert basis N of H and a unique quaternionic spectral measure F on the σ- algebra of \mathbb C_m⁺ (upper half plane of the slice complex plane \mathbb C_m) associated to T such that ⟨ x | Ty ⟩ = ∫_σ_S(T) ∩ ℂ_m⁺λ dF_x,y(λ), for all y ∈ D(T), x ∈ H, where F_x,y is a quaternion valued measure on the σ- algebra of ℂ_m⁺, for any x,y∈ H and σ_S(T) is the spherical spectrum of T. Here the representation of T is established with respect to the Hilbert basis N. To prove this result, we reduce the problem to the complex case and obtain the result by using the classical result.

Related

• The Spectral Theorem for Quaternionic Normal Operators
• Spectral Theorem for quaternionic normal operators: Multiplication form
• Normal operators in real and quaternionic Hilbert spaces
• Spectral properties of compact normal quaternionic operators
• The spectral theorem for quaternionic unbounded normal operators based on the S-spectrum