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Post Closed as "Not suitable for this site" by Michael Renardy, R.P., Yemon Choi, Christian Remling, Ben McKay
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Atransportconfusion
Atransportconfusion
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Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \ , \quad \mbox{ for any g} \in V \ . $$$$ \| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \leq 1 $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?

Given some operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \ , \quad \mbox{ for any g} \in V \ . $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?

Given some linear operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \sup_{g} \| Tg \|_W \ , \quad \mbox{ with } \| g \|_V \leq 1 $$

I am wondering whether given such an operator norm, and under certain conditions, we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?

added comma plus added functional analysis tag
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Atransportconfusion
Atransportconfusion
  • 11
  • 4

Given some operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \quad \mbox{ for any g} \in V \ . $$$$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \ , \quad \mbox{ for any g} \in V \ . $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?

Given some operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \quad \mbox{ for any g} \in V \ . $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?

Given some operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \ , \quad \mbox{ for any g} \in V \ . $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?

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Atransportconfusion
Atransportconfusion
  • 11
  • 4

Hierarchies of Operator Norms

Given some operator $T: V \mapsto W$, we can talk about the operator norm between the spaces V and W, i.e. $$ \| T \|_{V \mapsto W} \ = \ \| T g \|_W \quad \mbox{ for any g} \in V \ . $$

I am wondering whether given such an operator norm, and under certain conditions (e.g. $T$ is linear and/or the underlying domain is closed), we can say more general things about the operator norm between other spaces?

For example, if $V = W = L_2(0,1)$, then can we say anything about the norm $\| \cdot \|_{L_\infty \mapsto L_1}$ etc?