If a sequence of operator $A_n$ converges in norm to $A$, i.e. $\lim \lVert A_n-A\rVert=0$)where $A_n$ and $A\in B(H)$ ($H$ is the Hilbert space). Is it true that $A_n^*$ converges in norm to $A^*$?
The convergence of the adjoint operator
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$\begingroup$
functional-analysis
convergence-divergence
operator-theory
adjoint-operators
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0@DavideGiraudo Get it. – 2012-12-09
1 Answers
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Since $(A_n-A)^*=A_n^*-A^*$ and an operator has the same norm with the adjoint operator,so $||A_n-A||=||A_n^*-A^*||$,then we get the answer.
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0The step "an operator has the same norm as its adjoint" deserves more details. – 2012-12-09