Norm for adjoint action and relation to norm of matrix?
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Consider an operator or matrix $O$ the adjoint action then is $ad_O M=[O,M]$, where $M$ is an arbitrary test matrix.
My question as follows:
Is there a norm that has been defined for $ad_O$ and if so how does it relate to the operator norm of $O$? Take the Frobenius norm specificially $norm(O)=sqrtOO^dagger$. What norm for $ad_O$ is there that relates the easiest to this norm if any?
Pleas note that specificically I am looking for a relation of the form
$$leftlVert ad_OrightrVert_1=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$$,
where $f$ is a function and $g_n$ with $nin N$ are also functions.
Also I would like to reiterate that $ad$ is the adjoint action in this case and not the adjoint representation
matrices lie-algebras operator-algebras
asked Aug 22 at 6:24
Michael
63
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up vote
1
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Consider an operator or matrix $O$ the adjoint action then is $ad_O M=[O,M]$, where $M$ is an arbitrary test matrix.
My question as follows:
Is there a norm that has been defined for $ad_O$ and if so how does it relate to the operator norm of $O$? Take the Frobenius norm specificially $norm(O)=sqrtOO^dagger$. What norm for $ad_O$ is there that relates the easiest to this norm if any?
Pleas note that specificically I am looking for a relation of the form
$$leftlVert ad_OrightrVert_1=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$$,
where $f$ is a function and $g_n$ with $nin N$ are also functions.
Also I would like to reiterate that $ad$ is the adjoint action in this case and not the adjoint representation
matrices lie-algebras operator-algebras
asked Aug 22 at 6:24
Michael
63
Just a quick remark: The operator $O mapsto Ad_O$ that you mention has a kernel given by scalar diagonal matrices. You shall not expect an equality $| Ad_O |_1= | O |_2$, for two given norms, since the left hand side will be zero for those matrices and the right hand side not.
– Adrián González-Pérez
Aug 22 at 9:33
Oh that is not what I was asking. I was more interested in something of the form: $leftlVert ad_OrightrVert=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$. Also note that I was asking for $ad$ and not $Ad$
– Michael
Aug 23 at 4:54
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Consider an operator or matrix $O$ the adjoint action then is $ad_O M=[O,M]$, where $M$ is an arbitrary test matrix.
My question as follows:
Is there a norm that has been defined for $ad_O$ and if so how does it relate to the operator norm of $O$? Take the Frobenius norm specificially $norm(O)=sqrtOO^dagger$. What norm for $ad_O$ is there that relates the easiest to this norm if any?
Pleas note that specificically I am looking for a relation of the form
$$leftlVert ad_OrightrVert_1=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$$,
where $f$ is a function and $g_n$ with $nin N$ are also functions.
Also I would like to reiterate that $ad$ is the adjoint action in this case and not the adjoint representation
matrices lie-algebras operator-algebras
asked Aug 22 at 6:24
Michael
63
Consider an operator or matrix $O$ the adjoint action then is $ad_O M=[O,M]$, where $M$ is an arbitrary test matrix.
My question as follows:
Is there a norm that has been defined for $ad_O$ and if so how does it relate to the operator norm of $O$? Take the Frobenius norm specificially $norm(O)=sqrtOO^dagger$. What norm for $ad_O$ is there that relates the easiest to this norm if any?
Pleas note that specificically I am looking for a relation of the form
$$leftlVert ad_OrightrVert_1=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$$,
where $f$ is a function and $g_n$ with $nin N$ are also functions.
Also I would like to reiterate that $ad$ is the adjoint action in this case and not the adjoint representation
matrices lie-algebras operator-algebras
asked Aug 22 at 6:24
Michael
63
edited Aug 23 at 5:02
asked Aug 22 at 6:24
Michael
63
asked Aug 22 at 6:24
Michael
63
asked Aug 22 at 6:24
Michael
63
63
Just a quick remark: The operator $O mapsto Ad_O$ that you mention has a kernel given by scalar diagonal matrices. You shall not expect an equality $| Ad_O |_1= | O |_2$, for two given norms, since the left hand side will be zero for those matrices and the right hand side not.
– Adrián González-Pérez
Aug 22 at 9:33
Oh that is not what I was asking. I was more interested in something of the form: $leftlVert ad_OrightrVert=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$. Also note that I was asking for $ad$ and not $Ad$
– Michael
Aug 23 at 4:54
add a comment |Â
Just a quick remark: The operator $O mapsto Ad_O$ that you mention has a kernel given by scalar diagonal matrices. You shall not expect an equality $| Ad_O |_1= | O |_2$, for two given norms, since the left hand side will be zero for those matrices and the right hand side not.
– Adrián González-Pérez
Aug 22 at 9:33
Oh that is not what I was asking. I was more interested in something of the form: $leftlVert ad_OrightrVert=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$. Also note that I was asking for $ad$ and not $Ad$
– Michael
Aug 23 at 4:54
Just a quick remark: The operator $O mapsto Ad_O$ that you mention has a kernel given by scalar diagonal matrices. You shall not expect an equality $| Ad_O |_1= | O |_2$, for two given norms, since the left hand side will be zero for those matrices and the right hand side not.
– Adrián González-Pérez
Aug 22 at 9:33
Just a quick remark: The operator $O mapsto Ad_O$ that you mention has a kernel given by scalar diagonal matrices. You shall not expect an equality $| Ad_O |_1= | O |_2$, for two given norms, since the left hand side will be zero for those matrices and the right hand side not.
– Adrián González-Pérez
Aug 22 at 9:33
Oh that is not what I was asking. I was more interested in something of the form: $leftlVert ad_OrightrVert=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$. Also note that I was asking for $ad$ and not $Ad$
– Michael
Aug 23 at 4:54
Oh that is not what I was asking. I was more interested in something of the form: $leftlVert ad_OrightrVert=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$. Also note that I was asking for $ad$ and not $Ad$
– Michael
Aug 23 at 4:54
add a comment |Â
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Just a quick remark: The operator $O mapsto Ad_O$ that you mention has a kernel given by scalar diagonal matrices. You shall not expect an equality $| Ad_O |_1= | O |_2$, for two given norms, since the left hand side will be zero for those matrices and the right hand side not.
– Adrián González-Pérez
Aug 22 at 9:33
Oh that is not what I was asking. I was more interested in something of the form: $leftlVert ad_OrightrVert=f(leftlVert g_1(O)rightrVert,...,leftlVert g_n(O)rightrVert)$. Also note that I was asking for $ad$ and not $Ad$
– Michael
Aug 23 at 4:54