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‎I ‎have ‎recently started normal operations, but I face with problems in this ‎regard.‎‎ please guide ‎me.‎

a: ‎let‎ ‎$ T ‎\in K ( H ) $‎ ‎be‎ ‎normal, ‎show that ‎‎‎$‎T ‎\geq 0 ‎‎‎‎‎$ ‎if ‎only ‎if ‎all eigenvalues of the operator is equal to zero..‎‎

b:‎‎‎let‎ ‎$ T ‎\in K ( H ) $‎ ‎be‎ ‎$ T ‎\geq ‎0‎ $‎, ‎show ‎that‎ There ‎is a‎ ‎‎ positive and unique compact actuator ‎$ ‎A‎$‎ so ‎that‎ ‎$ ‎A‎^{2} =‎ ‎T‎. $‎

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    Have you seen the continuous functional calculus? – 2017-01-17
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    yes, I study about Functional calculus theorem for continuous functions, but i can not solve The above statement . – 2017-01-17
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    (a) looks wrong. – 2017-01-17
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    (a) is wrong. Any operator on a finite-dimensional space is compact, and many are positive (hence normal) and have nonzero spectrum. – 2017-01-17
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    What is the definition of a positive operator you are using? There are several equivalent starting places. – 2017-01-17
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    I suspect (a) should read "greater than or equal to zero". – 2017-01-18

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