Approximate normal operator using linear combination of orthogonal projections











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For every bounded normal operator $N in mathcal{B(H)}$, and every $epsilon>0$, there is a set ${P_n}$ of pairwise commuting orthogonal projections with sum $I$(identity operator) and corresponding set ${lambda_n}$ in $mathbb{C}$ s.t $||N-sum_{n}lambda_nP_n|| < epsilon$.



I was thinking using functional calculus for normal operator to prove it, but I am still unclear how to start it. Furthermore, there is another similar problem: for a bounded normal operator T such that $0leq Tleq I$, find a sequence of pairwise commuting projections s.t $T=sum_n frac{1}{2^n}P_n$, there is hint for this problem: Let $P_1=chi_{(1/2,1]}, P_2=chi_{(1/4,1/2]cup(3/4,1]}, dots$. I am not sure how to solve those two problems and their connections.



Any help will be appreciated.






functional-analysis operator-theory







share|cite|improve this question






















  • What is the functional calculus for you? For instance, if you already know that the smallest weak operator topology closed *-algebra containing $N$ is isometrically *-isomorphic to $L^infty$ of some measure space, then the two exercises boil down to approximating the identity function on the spectrum of $N$ by simple function.
    – Bartosz Malman
    Dec 5 at 18:31












  • @BartoszMalman I meant the continuous functional calculus for normal operator. Is it just by continuous functional calculus, then we just need to approximate the identity functions?
    –  apple
    Dec 5 at 19:44















up vote
1
down vote

favorite
1












For every bounded normal operator $N in mathcal{B(H)}$, and every $epsilon>0$, there is a set ${P_n}$ of pairwise commuting orthogonal projections with sum $I$(identity operator) and corresponding set ${lambda_n}$ in $mathbb{C}$ s.t $||N-sum_{n}lambda_nP_n|| < epsilon$.



I was thinking using functional calculus for normal operator to prove it, but I am still unclear how to start it. Furthermore, there is another similar problem: for a bounded normal operator T such that $0leq Tleq I$, find a sequence of pairwise commuting projections s.t $T=sum_n frac{1}{2^n}P_n$, there is hint for this problem: Let $P_1=chi_{(1/2,1]}, P_2=chi_{(1/4,1/2]cup(3/4,1]}, dots$. I am not sure how to solve those two problems and their connections.



Any help will be appreciated.






functional-analysis operator-theory







share|cite|improve this question






















  • What is the functional calculus for you? For instance, if you already know that the smallest weak operator topology closed *-algebra containing $N$ is isometrically *-isomorphic to $L^infty$ of some measure space, then the two exercises boil down to approximating the identity function on the spectrum of $N$ by simple function.
    – Bartosz Malman
    Dec 5 at 18:31












  • @BartoszMalman I meant the continuous functional calculus for normal operator. Is it just by continuous functional calculus, then we just need to approximate the identity functions?
    –  apple
    Dec 5 at 19:44













up vote
1
down vote

favorite
1









up vote
1
down vote

favorite
1






1





For every bounded normal operator $N in mathcal{B(H)}$, and every $epsilon>0$, there is a set ${P_n}$ of pairwise commuting orthogonal projections with sum $I$(identity operator) and corresponding set ${lambda_n}$ in $mathbb{C}$ s.t $||N-sum_{n}lambda_nP_n|| < epsilon$.



I was thinking using functional calculus for normal operator to prove it, but I am still unclear how to start it. Furthermore, there is another similar problem: for a bounded normal operator T such that $0leq Tleq I$, find a sequence of pairwise commuting projections s.t $T=sum_n frac{1}{2^n}P_n$, there is hint for this problem: Let $P_1=chi_{(1/2,1]}, P_2=chi_{(1/4,1/2]cup(3/4,1]}, dots$. I am not sure how to solve those two problems and their connections.



Any help will be appreciated.






functional-analysis operator-theory







share|cite|improve this question













For every bounded normal operator $N in mathcal{B(H)}$, and every $epsilon>0$, there is a set ${P_n}$ of pairwise commuting orthogonal projections with sum $I$(identity operator) and corresponding set ${lambda_n}$ in $mathbb{C}$ s.t $||N-sum_{n}lambda_nP_n|| < epsilon$.



I was thinking using functional calculus for normal operator to prove it, but I am still unclear how to start it. Furthermore, there is another similar problem: for a bounded normal operator T such that $0leq Tleq I$, find a sequence of pairwise commuting projections s.t $T=sum_n frac{1}{2^n}P_n$, there is hint for this problem: Let $P_1=chi_{(1/2,1]}, P_2=chi_{(1/4,1/2]cup(3/4,1]}, dots$. I am not sure how to solve those two problems and their connections.



Any help will be appreciated.






functional-analysis operator-theory




functional-analysis operator-theory






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 5 at 17:53









apple

467




467












  • What is the functional calculus for you? For instance, if you already know that the smallest weak operator topology closed *-algebra containing $N$ is isometrically *-isomorphic to $L^infty$ of some measure space, then the two exercises boil down to approximating the identity function on the spectrum of $N$ by simple function.
    – Bartosz Malman
    Dec 5 at 18:31












  • @BartoszMalman I meant the continuous functional calculus for normal operator. Is it just by continuous functional calculus, then we just need to approximate the identity functions?
    –  apple
    Dec 5 at 19:44


















  • What is the functional calculus for you? For instance, if you already know that the smallest weak operator topology closed *-algebra containing $N$ is isometrically *-isomorphic to $L^infty$ of some measure space, then the two exercises boil down to approximating the identity function on the spectrum of $N$ by simple function.
    – Bartosz Malman
    Dec 5 at 18:31












  • @BartoszMalman I meant the continuous functional calculus for normal operator. Is it just by continuous functional calculus, then we just need to approximate the identity functions?
    –  apple
    Dec 5 at 19:44
















What is the functional calculus for you? For instance, if you already know that the smallest weak operator topology closed *-algebra containing $N$ is isometrically *-isomorphic to $L^infty$ of some measure space, then the two exercises boil down to approximating the identity function on the spectrum of $N$ by simple function.
– Bartosz Malman
Dec 5 at 18:31






What is the functional calculus for you? For instance, if you already know that the smallest weak operator topology closed *-algebra containing $N$ is isometrically *-isomorphic to $L^infty$ of some measure space, then the two exercises boil down to approximating the identity function on the spectrum of $N$ by simple function.
– Bartosz Malman
Dec 5 at 18:31














@BartoszMalman I meant the continuous functional calculus for normal operator. Is it just by continuous functional calculus, then we just need to approximate the identity functions?
–  apple
Dec 5 at 19:44




@BartoszMalman I meant the continuous functional calculus for normal operator. Is it just by continuous functional calculus, then we just need to approximate the identity functions?
–  apple
Dec 5 at 19:44










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Using compactness of $sigma(N)$, find a partition $sigma(N)=bigcup_{n=1}^k K_n$ with each $K_n$ measurable and contained in a ball of radius $varepsilon$. Now let $P_n=1_{K_n}(N)$, and $lambda_nin K_n$. Then
begin{align}
|N-sum_nlambda_nP_n|&=left|sum_n (NP_n-lambda_nP_n)right|
=max_n|NP_n-lambda_nP_n| \
&=max_nleft|int_{K_n}(lambda-lambda_n),dE(lambda) right|
leqmax_nvarepsilon|P_n|=varepsilon.
end{align}



For the second part, you use the identity
$$
t=sum_{n=1}^infty 2^{-n},1_{R_n}(t),=sum_{n=1}^n2^{-n},1_{{tgeq 1-2^{-n}}}(t).
$$
where $R_n={t: tgeqsum_{k=1}^n2^{-k}}$.






share|cite|improve this answer





















  • are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
    –  apple
    Dec 5 at 19:45










  • Implicitly? I would say fairly explicitly.
    – Martin Argerami
    Dec 5 at 20:38











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accepted










Using compactness of $sigma(N)$, find a partition $sigma(N)=bigcup_{n=1}^k K_n$ with each $K_n$ measurable and contained in a ball of radius $varepsilon$. Now let $P_n=1_{K_n}(N)$, and $lambda_nin K_n$. Then
begin{align}
|N-sum_nlambda_nP_n|&=left|sum_n (NP_n-lambda_nP_n)right|
=max_n|NP_n-lambda_nP_n| \
&=max_nleft|int_{K_n}(lambda-lambda_n),dE(lambda) right|
leqmax_nvarepsilon|P_n|=varepsilon.
end{align}



For the second part, you use the identity
$$
t=sum_{n=1}^infty 2^{-n},1_{R_n}(t),=sum_{n=1}^n2^{-n},1_{{tgeq 1-2^{-n}}}(t).
$$
where $R_n={t: tgeqsum_{k=1}^n2^{-k}}$.






share|cite|improve this answer





















  • are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
    –  apple
    Dec 5 at 19:45










  • Implicitly? I would say fairly explicitly.
    – Martin Argerami
    Dec 5 at 20:38















up vote
0
down vote



accepted










Using compactness of $sigma(N)$, find a partition $sigma(N)=bigcup_{n=1}^k K_n$ with each $K_n$ measurable and contained in a ball of radius $varepsilon$. Now let $P_n=1_{K_n}(N)$, and $lambda_nin K_n$. Then
begin{align}
|N-sum_nlambda_nP_n|&=left|sum_n (NP_n-lambda_nP_n)right|
=max_n|NP_n-lambda_nP_n| \
&=max_nleft|int_{K_n}(lambda-lambda_n),dE(lambda) right|
leqmax_nvarepsilon|P_n|=varepsilon.
end{align}



For the second part, you use the identity
$$
t=sum_{n=1}^infty 2^{-n},1_{R_n}(t),=sum_{n=1}^n2^{-n},1_{{tgeq 1-2^{-n}}}(t).
$$
where $R_n={t: tgeqsum_{k=1}^n2^{-k}}$.






share|cite|improve this answer





















  • are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
    –  apple
    Dec 5 at 19:45










  • Implicitly? I would say fairly explicitly.
    – Martin Argerami
    Dec 5 at 20:38













up vote
0
down vote



accepted







up vote
0
down vote



accepted






Using compactness of $sigma(N)$, find a partition $sigma(N)=bigcup_{n=1}^k K_n$ with each $K_n$ measurable and contained in a ball of radius $varepsilon$. Now let $P_n=1_{K_n}(N)$, and $lambda_nin K_n$. Then
begin{align}
|N-sum_nlambda_nP_n|&=left|sum_n (NP_n-lambda_nP_n)right|
=max_n|NP_n-lambda_nP_n| \
&=max_nleft|int_{K_n}(lambda-lambda_n),dE(lambda) right|
leqmax_nvarepsilon|P_n|=varepsilon.
end{align}



For the second part, you use the identity
$$
t=sum_{n=1}^infty 2^{-n},1_{R_n}(t),=sum_{n=1}^n2^{-n},1_{{tgeq 1-2^{-n}}}(t).
$$
where $R_n={t: tgeqsum_{k=1}^n2^{-k}}$.






share|cite|improve this answer












Using compactness of $sigma(N)$, find a partition $sigma(N)=bigcup_{n=1}^k K_n$ with each $K_n$ measurable and contained in a ball of radius $varepsilon$. Now let $P_n=1_{K_n}(N)$, and $lambda_nin K_n$. Then
begin{align}
|N-sum_nlambda_nP_n|&=left|sum_n (NP_n-lambda_nP_n)right|
=max_n|NP_n-lambda_nP_n| \
&=max_nleft|int_{K_n}(lambda-lambda_n),dE(lambda) right|
leqmax_nvarepsilon|P_n|=varepsilon.
end{align}



For the second part, you use the identity
$$
t=sum_{n=1}^infty 2^{-n},1_{R_n}(t),=sum_{n=1}^n2^{-n},1_{{tgeq 1-2^{-n}}}(t).
$$
where $R_n={t: tgeqsum_{k=1}^n2^{-k}}$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 5 at 19:14









Martin Argerami

123k1176174




123k1176174












  • are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
    –  apple
    Dec 5 at 19:45










  • Implicitly? I would say fairly explicitly.
    – Martin Argerami
    Dec 5 at 20:38


















  • are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
    –  apple
    Dec 5 at 19:45










  • Implicitly? I would say fairly explicitly.
    – Martin Argerami
    Dec 5 at 20:38
















are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
–  apple
Dec 5 at 19:45




are you implicitly using the spectral theorem or continuous functional calculus for normal operators?
–  apple
Dec 5 at 19:45












Implicitly? I would say fairly explicitly.
– Martin Argerami
Dec 5 at 20:38




Implicitly? I would say fairly explicitly.
– Martin Argerami
Dec 5 at 20:38


















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