Calculate the operator norm of the difference of two operators
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Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
asked Nov 20 at 14:49
Max93
30028
add a comment |
up vote
0
down vote
favorite
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
asked Nov 20 at 14:49
Max93
30028
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 at 15:01
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
asked Nov 20 at 14:49
Max93
30028
Let $x_1,...,x_n in [0,1]$, $lambda_1,...,lambda_n in mathbb{C}$ and let $S,T : C([0,1]) to C([0,1])$ be the operators defined by
$T(g)= int_0^1 g(s), ds$
and
$S(g) = sum_{i=1}^n lambda_i g(x_i)$.
$C([0,1])$ is equipped with the $|cdot|_{infty}$-norm.
I already proved that $|T| = 1$ and $|S| = sum_{i=1}^n |lambda_i|$, where I used for the $geq $-inequality the function $tilde{g}$, which satisfies $tilde{g}(x_i) = dfrac{bar{lambda_i}}{|lambda_i|}$ for all $i=1,...,n$ and is linearly interpolated between the $x_i$.
Now I want to find the norm of the operator $R:=T-S$.
We obviously have
$|R| leq |T|+|S| = 1+sum_{i=1}^n |lambda_i|$
and
$|R| geq ||T|-|S|| = |1-sum_{i=1}^n |lambda_i||$.
How can I proceed now?
Thanks in advance!
functional-analysis operator-theory norm
functional-analysis operator-theory norm
asked Nov 20 at 14:49
Max93
30028
asked Nov 20 at 14:49
Max93
30028
asked Nov 20 at 14:49
Max93
30028
asked Nov 20 at 14:49
Max93
30028
asked Nov 20 at 14:49
Max93
30028
30028
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 at 15:01
add a comment |
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 at 15:01
1
1
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 at 15:01
The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 at 15:01
add a comment |
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The norm of $R$ is equal to the upper bound you calculated: define a continuous function which is $pm 1$ at $x_i$ and equal to $1$ at point with distance $epsilon$ to all $x_i$. Then let $epsilonsearrow0$.
– daw
Nov 20 at 15:01