Operator norm

In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ${\displaystyle \|T\|}$ of a linear map ${\displaystyle T:X\to Y}$ is the maximum factor by which it "lengthens" vectors.

Introduction and definition

Given two normed vector spaces ${\displaystyle V}$  and ${\displaystyle W}$  (over the same base field, either the real numbers ${\displaystyle \mathbb {R} }$  or the complex numbers ${\displaystyle \mathbb {C} }$ ), a linear map ${\displaystyle A:V\to W}$  is continuous if and only if there exists a real number ${\displaystyle c}$  such that^[1]

$${\displaystyle \|Av\|\leq c\|v\|\quad {\mbox{ for all }}v\in V.}$$

 

The norm on the left is the one in ${\displaystyle W}$  and the norm on the right is the one in ${\displaystyle V}$ . Intuitively, the continuous operator ${\displaystyle A}$  never increases the length of any vector by more than a factor of ${\displaystyle c.}$  Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of ${\displaystyle A,}$  one can take the infimum of the numbers ${\displaystyle c}$  such that the above inequality holds for all ${\displaystyle v\in V.}$  This number represents the maximum scalar factor by which ${\displaystyle A}$  "lengthens" vectors. In other words, the "size" of ${\displaystyle A}$  is measured by how much it "lengthens" vectors in the "biggest" case. So we define the operator norm of ${\displaystyle A}$  as

$${\displaystyle \|A\|_{op}=\inf\{c\geq 0:\|Av\|\leq c\|v\|{\mbox{ for all }}v\in V\}.}$$

 

The infimum is attained as the set of all such ${\displaystyle c}$  is closed, nonempty, and bounded from below.^[2]

It is important to bear in mind that this operator norm depends on the choice of norms for the normed vector spaces ${\displaystyle V}$  and ${\displaystyle W}$ .

Examples

Every real ${\displaystyle m}$ -by-${\displaystyle n}$  matrix corresponds to a linear map from ${\displaystyle \mathbb {R} ^{n}}$  to ${\displaystyle \mathbb {R} ^{m}.}$  Each pair of the plethora of (vector) norms applicable to real vector spaces induces an operator norm for all ${\displaystyle m}$ -by-${\displaystyle n}$  matrices of real numbers; these induced norms form a subset of matrix norms.

If we specifically choose the Euclidean norm on both ${\displaystyle \mathbb {R} ^{n}}$  and ${\displaystyle \mathbb {R} ^{m},}$  then the matrix norm given to a matrix ${\displaystyle A}$  is the square root of the largest eigenvalue of the matrix ${\displaystyle A^{*}A}$  (where ${\displaystyle A^{*}}$  denotes the conjugate transpose of ${\displaystyle A}$ ).^[3] This is equivalent to assigning the largest singular value of ${\displaystyle A.}$ 

Passing to a typical infinite-dimensional example, consider the sequence space ${\displaystyle \ell ^{2},}$  which is an L^p space, defined by

$${\displaystyle l^{2}=\left\{\left(a_{n}\right)_{n\geq 1}:\;a_{n}\in \mathbb {C} ,\;\sum _{n}|a_{n}|^{2}<\infty \right\}.}$$

 

This can be viewed as an infinite-dimensional analogue of the Euclidean space ${\displaystyle \mathbb {C} ^{n}.}$  Now consider a bounded sequence ${\displaystyle s_{\bullet }=\left(s_{n}\right)_{n=1}^{\infty }.}$  The sequence ${\displaystyle s_{\bullet }}$  is an element of the space ${\displaystyle \ell ^{\infty },}$  with a norm given by

$${\displaystyle \left\|s_{\bullet }\right\|_{\infty }=\sup _{n}\left|s_{n}\right|.}$$

 

Define an operator ${\displaystyle T_{s}}$  by pointwise multiplication:

$${\displaystyle \left(a_{n}\right)_{n=1}^{\infty }\;{\stackrel {T_{s}}{\mapsto }}\;\ \left(s_{n}\cdot a_{n}\right)_{n=1}^{\infty }.}$$

 

The operator ${\displaystyle T_{s}}$  is bounded with operator norm

$${\displaystyle \left\|T_{s}\right\|_{op}=\left\|s_{\bullet }\right\|_{\infty }.}$$

 

This discussion extends directly to the case where ${\displaystyle \ell ^{2}}$  is replaced by a general ${\displaystyle L^{p}}$  space with ${\displaystyle p>1}$  and ${\displaystyle \ell ^{\infty }}$  replaced by ${\displaystyle L^{\infty }.}$ 

Equivalent definitions

Let ${\displaystyle A:V\to W}$  be a linear operator between normed spaces. The first four definitions are always equivalent, and if in addition ${\displaystyle V\neq \{0\}}$  then they are all equivalent:

${\displaystyle {\begin{alignedat}{4}\|A\|_{op}&=\inf &&\{c\geq 0~&&:~\|Av\|\leq c\|v\|~&&~{\mbox{ for all }}~&&v\in V\}\\&=\sup &&\{\|Av\|~&&:~\|v\|\leq 1~&&~{\mbox{ and }}~&&v\in V\}\\&=\sup &&\{\|Av\|~&&:~\|v\|<1~&&~{\mbox{ and }}~&&v\in V\}\\&=\sup &&\{\|Av\|~&&:~\|v\|\in \{0,1\}~&&~{\mbox{ and }}~&&v\in V\}\\&=\sup &&\{\|Av\|~&&:~\|v\|=1~&&~{\mbox{ and }}~&&v\in V\}\;\;\;{\text{ this equality holds if and only if }}V\neq \{0\}\\&=\sup &&{\bigg \{}{\frac {\|Av\|}{\|v\|}}~&&:~v\neq 0~&&~{\mbox{ and }}~&&v\in V{\bigg \}}\;\;\;{\text{ this equality holds if and only if }}V\neq \{0\}.\\\end{alignedat}}}$ 

If ${\displaystyle V=\{0\}}$  then the sets in the last two rows will be empty, and consequently their supremums over the set ${\displaystyle [-\infty ,\infty ]}$  will equal ${\displaystyle -\infty }$  instead of the correct value of ${\displaystyle 0.}$  If the supremum is taken over the set ${\displaystyle [0,\infty ]}$  instead, then the supremum of the empty set is ${\displaystyle 0}$  and the formulas hold for any ${\displaystyle V.}$ 

Importantly, a linear operator ${\displaystyle A:V\to W}$  is not, in general, guaranteed to achieve its norm ${\displaystyle \|A\|_{op}=\sup\{\|Av\|:\|v\|\leq 1,v\in V\}}$  on the closed unit ball ${\displaystyle \{v\in V:\|v\|\leq 1\},}$  meaning that there might not exist any vector ${\displaystyle u\in V}$  of norm ${\displaystyle \|u\|\leq 1}$  such that ${\displaystyle \|A\|_{op}=\|Au\|}$  (if such a vector does exist and if ${\displaystyle A\neq 0,}$  then ${\displaystyle u}$  would necessarily have unit norm ${\displaystyle \|u\|=1}$ ). R.C. James proved James's theorem in 1964, which states that a Banach space ${\displaystyle V}$  is reflexive if and only if every bounded linear functional ${\displaystyle f\in V^{*}}$  achieves its norm on the closed unit ball.^[4] It follows, in particular, that every non-reflexive Banach space has some bounded linear functional (a type of bounded linear operator) that does not achieve its norm on the closed unit ball.

If ${\displaystyle A:V\to W}$  is bounded then^[5]

$${\displaystyle \|A\|_{op}=\sup \left\{\left|w^{*}(Av)\right|:\|v\|\leq 1,\left\|w^{*}\right\|\leq 1{\text{ where }}v\in V,w^{*}\in W^{*}\right\}}$$

 

and^[5]

$${\displaystyle \|A\|_{op}=\left\|{}^{t}A\right\|_{op}}$$

 

where ${\displaystyle {}^{t}A:W^{*}\to V^{*}}$  is the transpose of ${\displaystyle A:V\to W,}$  which is the linear operator defined by ${\displaystyle w^{*}\,\mapsto \,w^{*}\circ A.}$ 

Properties

The operator norm is indeed a norm on the space of all bounded operators between ${\displaystyle V}$  and ${\displaystyle W}$ . This means

$${\displaystyle \|A\|_{op}\geq 0{\mbox{ and }}\|A\|_{op}=0{\mbox{ if and only if }}A=0,}$$

 

$${\displaystyle \|aA\|_{op}=|a|\|A\|_{op}{\mbox{ for every scalar }}a,}$$

 

$${\displaystyle \|A+B\|_{op}\leq \|A\|_{op}+\|B\|_{op}.}$$

 

The following inequality is an immediate consequence of the definition:

$${\displaystyle \|Av\|\leq \|A\|_{op}\|v\|\ {\mbox{ for every }}\ v\in V.}$$

 

The operator norm is also compatible with the composition, or multiplication, of operators: if ${\displaystyle V}$ , ${\displaystyle W}$  and ${\displaystyle X}$  are three normed spaces over the same base field, and ${\displaystyle A:V\to W}$  and ${\displaystyle B:W\to X}$  are two bounded operators, then it is a sub-multiplicative norm, that is:

$${\displaystyle \|BA\|_{op}\leq \|B\|_{op}\|A\|_{op}.}$$

 

For bounded operators on ${\displaystyle V}$ , this implies that operator multiplication is jointly continuous.

It follows from the definition that if a sequence of operators converges in operator norm, it converges uniformly on bounded sets.

Table of common operator norms

By choosing different norms for the codomain, used in computing ${\displaystyle \|Av\|}$ , and the domain, used in computing ${\displaystyle \|v\|}$ , we obtain different values for the operator norm. Some common operator norms are easy to calculate, and others are NP-hard. Except for the NP-hard norms, all these norms can be calculated in ${\displaystyle N^{2}}$  operations (for an ${\displaystyle N\times N}$  matrix), with the exception of the ${\displaystyle \ell _{2}-\ell _{2}}$  norm (which requires ${\displaystyle N^{3}}$  operations for the exact answer, or fewer if you approximate it with the power method or Lanczos iterations).

Computability of Operator Norms^[6]

		Co-domain
		${\displaystyle \ell _{1}}$	${\displaystyle \ell _{2}}$	${\displaystyle \ell _{\infty }}$
Domain	${\displaystyle \ell _{1}}$	Maximum ${\displaystyle \ell _{1}}$  norm of a column	Maximum ${\displaystyle \ell _{2}}$  norm of a column	Maximum ${\displaystyle \ell _{\infty }}$  norm of a column
	${\displaystyle \ell _{2}}$	NP-hard	Maximum singular value	Maximum ${\displaystyle \ell _{2}}$  norm of a row
	${\displaystyle \ell _{\infty }}$	NP-hard	NP-hard	Maximum ${\displaystyle \ell _{1}}$  norm of a row

The norm of the adjoint or transpose can be computed as follows. We have that for any ${\displaystyle p,q,}$  then ${\displaystyle \|A\|_{p\rightarrow q}=\|A^{*}\|_{q'\rightarrow p'}}$  where ${\displaystyle p',q'}$  are Hölder conjugate to ${\displaystyle p,q,}$  that is, ${\displaystyle 1/p+1/p'=1}$  and ${\displaystyle 1/q+1/q'=1.}$ 

Operators on a Hilbert space

Suppose ${\displaystyle H}$  is a real or complex Hilbert space. If ${\displaystyle A:H\to H}$  is a bounded linear operator, then we have

$${\displaystyle \|A\|_{op}=\left\|A^{*}\right\|_{op}}$$

 

and

$${\displaystyle \left\|A^{*}A\right\|_{op}=\|A\|_{op}^{2},}$$

 

where ${\displaystyle A^{*}}$  denotes the adjoint operator of ${\displaystyle A}$  (which in Euclidean spaces with the standard inner product corresponds to the conjugate transpose of the matrix ${\displaystyle A}$ ).

In general, the spectral radius of ${\displaystyle A}$  is bounded above by the operator norm of ${\displaystyle A}$ :

$${\displaystyle \rho (A)\leq \|A\|_{op}.}$$

 

To see why equality may not always hold, consider the Jordan canonical form of a matrix in the finite-dimensional case. Because there are non-zero entries on the superdiagonal, equality may be violated. The quasinilpotent operators is one class of such examples. A nonzero quasinilpotent operator ${\displaystyle A}$  has spectrum ${\displaystyle \{0\}.}$  So ${\displaystyle \rho (A)=0}$  while ${\displaystyle \|A\|_{op}>0.}$ 

However, when a matrix ${\displaystyle N}$  is normal, its Jordan canonical form is diagonal (up to unitary equivalence); this is the spectral theorem. In that case it is easy to see that

$${\displaystyle \rho (N)=\|N\|_{op}.}$$

 

This formula can sometimes be used to compute the operator norm of a given bounded operator ${\displaystyle A}$ : define the Hermitian operator ${\displaystyle B=A^{*}A,}$  determine its spectral radius, and take the square root to obtain the operator norm of ${\displaystyle A.}$ 

The space of bounded operators on ${\displaystyle H,}$  with the topology induced by operator norm, is not separable. For example, consider the Lp space ${\displaystyle L^{2}[0,1],}$  which is a Hilbert space. For ${\displaystyle 0<t\leq 1,}$  let ${\displaystyle \Omega _{t}}$  be the characteristic function of ${\displaystyle [0,t],}$  and ${\displaystyle P_{t}}$  be the multiplication operator given by ${\displaystyle \Omega _{t},}$  that is,

$${\displaystyle P_{t}(f)=f\cdot \Omega _{t}.}$$

 

Then each ${\displaystyle P_{t}}$  is a bounded operator with operator norm 1 and

$${\displaystyle \left\|P_{t}-P_{s}\right\|_{op}=1\quad {\mbox{ for all }}\quad t\neq s.}$$

 

But ${\displaystyle \{P_{t}:0<t\leq 1\}}$  is an uncountable set. This implies the space of bounded operators on ${\displaystyle L^{2}([0,1])}$  is not separable, in operator norm. One can compare this with the fact that the sequence space ${\displaystyle \ell ^{\infty }}$  is not separable.

The associative algebra of all bounded operators on a Hilbert space, together with the operator norm and the adjoint operation, yields a C*-algebra.

See also

• Banach–Mazur compactum – Set of n-dimensional subspaces of a normed space made into a compact metric space.
• Continuous linear operator
• Contraction (operator theory) – Bounded operators with sub-unit norm
• Discontinuous linear map
• Dual norm – Measurement on a normed vector space
• Matrix norm – Norm on a vector space of matrices
• Norm (mathematics) – Length in a vector space
• Normed space – Vector space on which a distance is definedPages displaying short descriptions of redirect targets
• Operator algebra – Branch of functional analysis
• Operator theory – Mathematical field of study
• Topologies on the set of operators on a Hilbert space
• Unbounded operator – Linear operator defined on a dense linear subspace

Notes

1. Kreyszig, Erwin (1978), Introductory functional analysis with applications, John Wiley & Sons, p. 97, ISBN 9971-51-381-1
2. See e.g. Lemma 6.2 of Aliprantis & Border (2007).
3. Weisstein, Eric W. "Operator Norm". mathworld.wolfram.com. Retrieved 2020-03-14.
4. Diestel 1984, p. 6.
5. Rudin 1991, pp. 92–115.
6. section 4.3.1, Joel Tropp's PhD thesis, [1]

References

• Aliprantis, Charalambos D.; Border, Kim C. (2007), Infinite Dimensional Analysis: A Hitchhiker's Guide, Springer, p. 229, ISBN 9783540326960.
• Conway, John B. (1990), "III.2 Linear Operators on Normed Spaces", A Course in Functional Analysis, New York: Springer-Verlag, pp. 67–69, ISBN 0-387-97245-5
• Diestel, Joe (1984). Sequences and series in Banach spaces. New York: Springer-Verlag. ISBN 0-387-90859-5. OCLC 9556781.
• Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277.