I understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?
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$\begingroup$The operator norm is a norm defined on the space of bounded linear operators between two given normed vector spaces $X$ & $Y.$ Informally, the operator norm is a method by which we can measure the “size” of a given linear operator.
Let $X$ & $Y$ be two normed spaces. Define a continuous linear map as $A:X\to Y$ satisfying
\begin{equation*} ||Ax||\leq c||x||~\forall x\in X,~c\in\mathbb{R}. \end{equation*}
To understand operator norms, think of the continuous operator $A$ as never lengthening a vector by more than a factor of $c.$
Continuous linear operators are called bounded operators because the image of a bounded set under a continuous operator is also bounded.
If we want to measure the size of $A$ then what seems intuitive is to take the smallest number $c$ such that our inequality holds $\forall x\in X.$ ie, we measure the size of $A$ by how much it lengthens vectors in the biggest case.
Therefore we can define the operator norm of $A$ as the set
\begin{equation*} \inf\{ c\geq 0||Ax||\leq c||x||~\forall x\in X\}. \end{equation*}
I am interested in the norms of certain operators. For example, define the Hilbert transform (in singular integral form)
\begin{equation*} Hf(x)=\frac{1}{\pi}\int_{\mathbb{R}}\frac{f(y)}{x-y}dy. \end{equation*}
This is a linear operator. Pichorides theorem gives the $L^p$ bound of the Hilbert transform as
\begin{equation*} ||H||_p=\cot(\frac{\pi}{2p^*}) \end{equation*}
where $p^*=\max(p,p’),~\frac{1}{p}+\frac{1}{p’}=1.$
Now define the Riesz transform (generalization of the Hilbert transform)
\begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. \end{equation*}
The Iwaniec-Martin theorem gives the $L^p$ bound as
\begin{equation*} ||R_j||_p=\cot(\frac{\pi}{2p^*}). \end{equation*}
Define the Beurling-Ahlfors transform as
\begin{equation*} Sf(z)=-\frac{1}{\pi}\int_{\mathbb{C}}\frac{f(\zeta)}{(z-\zeta)^2}|d\zeta|^2. \end{equation*}
We cannot find the operator norm for this on $L^p$ because it has an even kernel so the usual methods for checking $L^p$ boundedness (such as the method of rotations that only works for singular integrals with odd kernels cannot be applied). This leads to the Iwaniec conjecture which states $||S||_p=(p^*-1).$
I hope this helps.
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