nLab
bounded operator

Context

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Functional analysis

AQFT

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory: classical, pre-quantum, quantum, perturbative quantum

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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

Last revised on February 7, 2017 at 12:32:00. See the history of this page for a list of all contributions to it.