Von Neumann inequality

**Drexel28** · January 11th 2010, 01:38 PM

Hello friends.

A friend of mine is doing research with a professor on the 3 by 3 multivariable case of the Von Neumann inequality. I am helping him out a little bit, but we are running into some techincal problems. For example, what is the quickest way to calculate the operator norm of a contraction matrix? For example what is $\|A\|$ where $A=\frac{1}{2}I_3$ ?

**Jose27** · January 11th 2010, 02:53 PM

Originally Posted by Drexel28

Hello friends.

A friend of mine is doing research with a professor on the 3 by 3 multivariable case of the Von Neumann inequality. I am helping him out a little bit, but we are running into some techincal problems. For example, what is the quickest way to calculate the operator norm of a contraction matrix? For example what is $\|A\|$ where $A=\frac{1}{2}I_3$ ?

For your example the usual definition works fine since $\Vert Av \Vert = 1/2$ for all $v\in \mathbb{S} ^2$ . For diagonal matices (or any matrix for which $A^*A$ is simple enough) you could use $\Vert A \Vert = \sqrt{ \lambda _{max} (A^*A) }$ where $\lambda _{max} (A^*A)$ is the largest eigenvaule of $A^*A$ and $A^*$ means the adjoint of $A$ .

**Moo** · January 11th 2010, 02:54 PM

Hello,

Like for the vectors, there are many norms for a matrix...

In general, we define $\|A\|_p=\max_{\|x\|_p\neq 0} ~\frac{\|Ax\|_p}{\|x\|_p}=\max_{\|x\|_p=1} ~\|Ax\|_p$

where $\|(x_1,\dots,x_n)\|_p=\left(\sum_{i=1}^n |x_i|^p\right)^{\frac 1p}$

This is easy to deal with if A is nice... Otherwise, there are a few results (that can be proved) :

$\|A\|_\infty=\max_{1\leq i\leq n} \sum_{j=1}^n |a_{ij}|$

$\|A\|_1=\max_{1\leq j\leq n} \sum_{i=1}^n |a_{ij}|$

$\|A\|_2=\sqrt{\rho(A'A)}$ , where $\rho(M)=\max ~|\lambda_i|$ ( $\lambda_i$ are the eigenvalues of M)

**Rebesques** · January 13th 2010, 12:41 PM

For example what is $\|A\|$ where $A=\frac{1}{2}I_3$ ?

1/2. I took a uni course once, where the professor would throw students out of the classroom for asking something like this.

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