Talk:Operator norm

Some rewriting needed?

I wonder if this article needs some rewriting.

First, the article starts with a very technical "all operator norm definitions money can buy", which is certainly not encouraging.

Also, the first section does not tie in well with the remainder of the article in general.

Looking down below, one uses the words "bounded linear transformation" to introduce the notion of norm. Some care is needed here, as a bounded linear operator is not implying it to be a bounded linear function.

What could maybe make this article better is to split it into two articles. One main one about linear bounded operators where one focuses on linear operators and not as much on fine details about their norms, and a shorter article on operator norms, which could be more techical.

Other ideas? Oleg Alexandrov 03:30, 21 Feb 2005 (UTC)

I agree, really. The edit from last June which introduced the definition by symbols is not really in the right place. It should go after the more verbose definition. Also the first paragraph should be expanded with some gentle introduction to the general idea: the norm of an operator gives us a specific way to talk about its 'size', operator norms apply to matrices but also are particularly useful in picturing the case of spaces of infinite dimension, if the operator norm fails to be defined because the sup is unbounded that indicates the operator is not continuous.

Charles Matthews 09:15, 21 Feb 2005 (UTC)

Ah, I see that bounded operator redirects here. Well, that is a self-link that needs to be fixed. So, Oleg's idea maybe to split that into a separate article seems quite promising. Charles Matthews 09:18, 21 Feb 2005 (UTC)

I will get to it by this weeked. It would be not easy to separate linear operator from operator norm, as these go hand in hand. However, the way things are now is not so good, so two articles, even with a small amount of repetition among them, should be better. Oleg Alexandrov 19:05, 21 Feb 2005 (UTC)

Infimum and minimum

Just a minor thing: the second section says that $\{c : \|Av\| \leq c\|v\| \forall v \in V\}$ may have no minimum, but I dispute that. It's the intersection over all nonzero $v\in V$ of the closed set $[\|Av\| / \|v\|, \infty]$, and an abritrary intersection of closed sets is closed. David Bulger 00:59, 12 May 2006 (UTC)[reply]

You are right, I reworded the text appropriately. Oleg Alexandrov (talk) 02:54, 12 May 2006 (UTC)[reply]

Inconsistency between operator norm and induced norm article

In the section about induced norms it says that the induced norm is the same as the operator norm. Then one definition is given, which is not the same definition given in this article about operator norms. This is a bit confusing. Is it possible to coordinate this information so both of the articles give the same information? Also, the information given in the latter article, does not seem to correspond to the information given in the WolframMathWorld articles about Operator norm and Induced norm. Mårten Berglund (talk) 15:07, 12 September 2010 (UTC)[reply]

Table of common operator norms

Some common operator norms are easy to calculate, and others are NP-hard. There's a very nice table in section 4.3.1 of Joel Tropp's PhD thesis from 2004[1]. I think this might make a nice addition to the main article. (This could be made more complete by discussing which of the NP-hard norms allow quick approximations).

Except for the NP-hard norms, all these norms can be calculated in N^2 operations (for a NxN matrix), with the exception of the l2-l2 norm (which requires N^3 operations for the exact answer, or less if you approximate it with the power method or Lanczos iterations).

Computability of Operator Norm
	Co-domain
	×	$\ell _{1}$	$\ell _{2}$	$\ell _{\infty }$
Domain	$\ell _{1}$	Maximum $\ell _{1}$ norm of a column	Maximum $\ell _{2}$ of a column	Maximum absolute entry of matrix
	$\ell _{2}$	NP-hard	Maximum singular value	Maximum $\ell _{2}$ of a row
	$\ell _{\infty }$	NP-hard	NP-hard	Maximum $\ell _{1}$ norm of a row

Lavaka (talk) 14:55, 20 April 2012 (UTC)[reply]

Error in "Operators on a Hilbert space" section?

Maybe I'm missing something, but this passage seems wrong:

Suppose H is a real or complex Hilbert space. If A : H → H is a bounded linear operator, then we have

\|A\|_{op}=\|A^{*}\|_{op}

If I choose H= R² with standard inner product and $A={\begin{bmatrix}1&2\\3&4\\\end{bmatrix}},$ then $\|A\|_{1}=6$ , but $\|A^{*}\|_{1}=7$ .

Sorry for any mistakes and thanks for any clarifications.

Saung Tadashi (talk) 21:25, 12 May 2012 (UTC)[reply]

I think the resolution is that your example isn't in Hilbert space! With H= R² with standard inner product, then we're forced to use the 2 norm, not the 1 norm. 128.138.65.148 (talk) 22:44, 29 March 2022 (UTC)[reply]

Ah, that's true! Thanks for clarifying :)

Saung Tadashi (talk) 07:58, 30 March 2022 (UTC)[reply]