operator norm

Definition

Let $A:𝖵\to 𝖶$ be a linear map between normed vector spaces $𝖵$ and $𝖶$. To each such map (operator) $A$ we can assign a non-negative number ${\parallel A\parallel }_{\mathrm{op}}$ defined by

$${\parallel A\parallel }_{\mathrm{op}}:=\underset{𝐯\in 𝖵}{sup}\frac{\parallel A𝐯\parallel }{\parallel 𝐯\parallel },$$

where the supremum ${\parallel A\parallel }_{\mathrm{op}}$ could be finite or infinite. Equivalently, the above definition can be written as

By convention, if $𝖵$ is the zero vector space, any operator from $𝖵$ to $𝖶$ must be the zero operator and is assigned zero norm.

${\parallel A\parallel }_{\mathrm{op}}$ is called the the operator norm (or the induced norm) of $A$, for reasons that will be clear in the next .

Operator norm is in fact a norm

Definition - If ${\parallel A\parallel }_{\mathrm{op}}$ is finite, we say that $A$ is a . Otherwise, we say that $A$ is .

It turns out that, for bounded operators, $\parallel \cdot {\parallel }_{\mathrm{op}}$ satisfies all the properties of a norm (hence the name operator norm). The proof follows immediately from the definition:

Positivity:

Since $\parallel A𝐯\parallel \ge 0$, by definition ${\parallel A\parallel }_{\mathrm{op}}\ge 0$. Also, $\parallel A𝐯\parallel =0$ identically only if $A=0$. Hence ${\parallel A\parallel }_{\mathrm{op}}=0$ only if $A=0$.

Absolute homogeneity:

Since $\parallel \lambda A𝐯\parallel =|\lambda |\parallel A𝐯\parallel$, by definition ${\parallel \lambda A\parallel }_{\mathrm{op}}=|\lambda |{\parallel A\parallel }_{\mathrm{op}}$.

Triangle inequality:

Since $\parallel (A+B)𝐯\parallel =\parallel A𝐯+B𝐯\parallel \le \parallel A𝐯\parallel +\parallel B𝐯\parallel$, by definition ${\parallel A+B\parallel }_{\mathrm{op}}\le {\parallel A\parallel }_{\mathrm{op}}+{\parallel B\parallel }_{\mathrm{op}}$.

The set $L(𝖵,𝖶)$ of bounded linear maps from $𝖵$ to $𝖶$ forms a vector space and $\parallel \cdot {\parallel }_{\mathrm{op}}$ defines a norm in it.

Example

Suppose that $𝖵=({ℝ}^n,\parallel \cdot {\parallel }_p)$ and $𝖶=({ℝ}^n,\parallel \cdot {\parallel }_p)$, where $\parallel \cdot {\parallel }_p$ is the vector p-norm. Then the operator norm $\parallel \cdot {\parallel }_{\mathrm{op}}=\parallel \cdot {\parallel }_p$ is the matrix p-norm.

Title	operator norm
Canonical name	OperatorNorm
Date of creation	2013-03-22 12:43:20
Last modified on	2013-03-22 12:43:20
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	15
Author	asteroid (17536)
Entry type	Definition
Classification	msc 47L25
Classification	msc 46A32
Classification	msc 47A30
Synonym	induced norm
Related topic	VectorNorm
Related topic	OperatorTopologies
Related topic	HomomorphismsOfCAlgebrasAreContinuous
Related topic	CAlgebra
Defines	bounded linear map
Defines	unbounded linear map
Defines	bounded operator
Defines	unbounded operator