nLab
bounded operator

Context

Operator algebra

Functional analysis

AQFT

Contents

Idea

In a (Hausdorff) topological vector space one can consider bounded sets: a set SS is bounded if it is absorbed by any open neighborhood UU of zero (i.e. a dilated multiple λU={λx|xU}\lambda U = \{\lambda x | x\in U\} contains SS). This specializes to the usual definition of a bounded set in a normed vector space: a set is bounded if it is contained in a ball of some finite radius r>0r \gt 0.

A linear operator A:V 1V 2A: V_1\to V_2 between topological vector spaces is bounded if it sends each bounded set in V 1V_1 to a bounded set in V 2V_2. For normed spaces, this is equivalent to saying that it sends the unit ball to a bounded set. Between finite dimensional normed spaces, every linear operator is bounded. A linear operator between any two normed linear spaces is bounded iff it is continuous.

There is also a rich theory for unbounded operators on Hilbert spaces.

The quotient of the bounded operators by the compact operators is called the Calkin algebra.

Properties

Every bounded operator on a Hilbert space has a polar decomposition.

The Gelfand-Naimark theorem says that every C*-algebra is isomorphic to a C *C^\ast-algebra of bounded linear operators on a Hilbert space.

Examples

Important classes of bounded operators are the compact operators, trace-class operators and Hilbert-Schmidt operators.

References

category: analysis

Revised on February 7, 2017 12:32:00 by Urs Schreiber (46.183.103.17)