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Suppose V is a complex inner product space. Prove that T E L(V) is normal if and only if there exists commuting self-adjoint operators S1, S₂ € L(V) such that T = S₁ +iS₂.

Suppose V is a complex inner product space. Prove that T E L(V) is normal if and only if there exists commuting self-adjoint operators S1, S₂ € L(V) such that T = S₁ +iS₂.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.2: Ring Homomorphisms
Problem 6E
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Suppose V is a complex inner product space. Prove that T¤ L(V) is normal if and only if there exists commuting self-adjoint operators S₁, S₂ € L(V) such that T = S₁ + iS₂. (Hint: Any T ≤ L(V) can be written as (T+T*) + i ½ (T − T*).) 2i
Transcribed Image Text:Suppose V is a complex inner product space. Prove that T¤ L(V) is normal if and only if there exists commuting self-adjoint operators S₁, S₂ € L(V) such that T = S₁ + iS₂. (Hint: Any T ≤ L(V) can be written as (T+T*) + i ½ (T − T*).) 2i
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