In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces is the maximum factor by which it "lengthens" vectors. It is also called the bound norm.

Introduction and definition

(over the same base field, either the real numbers or the complex numbers), a linear map is continuous if and only if such that

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 one can take the infimum

is closed, nonempty, and bounded from below.

.

Examples

matrix Each pair of the plethora of (vector) norms matrices of real numbers; these induced norms form a subset of matrix norms.

If we specifically choose the Euclidean norm is the square root of the largest eigenvalue denotes the conjugate transpose). This is equivalent to assigning the largest singular value

Passing to a typical infinite-dimensional example, consider the sequence space which is an Lp space

This can be viewed as an infinite-dimensional analogue of the Euclidean space

Equivalent definitions

then they are all equivalent:

then the sets in the last two rows will be empty, and consequently their supremums

). R.C. James proved James's theorem in 1964, which states that a Banach space is reflexive if and only if every bounded linear functional achieves its norm on the closed unit ball. 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.

is bounded then and is the transpose

Properties

The operator norm is indeed a norm on the space of all bounded operators

are two bounded operators, then it is a sub-multiplicative norm

, 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

, we obtain different values for the operator norm. Some common operator norms are easy to calculate, and others are NP-hard operations for the exact answer, or fewer if you approximate it with the power method or Lanczos iterations).The norm of the adjoint are Hölder conjugate

Operators on a Hilbert space

is a real or complex Hilbert space denotes the adjoint operator (which in Euclidean spaces with the standard inner product corresponds to the conjugate transpose).

In general, the spectral radius

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 normal, its Jordan canonical form is diagonal (up to unitary equivalence); this is the spectral theorem

: define the Hermitian operator determine its spectral radius, and take the square root

with the topology induced by operator norm, is not separable. For example, consider the Lp space be the characteristic function be the multiplication operator

is an uncountable set 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

Notes

References

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