Why is the domain of an unbounded operator so important?

An unbounded operator cannot act on every vector, so it is defined only on a dense subspace; the choice of that domain determines whether the operator is self-adjoint and therefore whether the spectral theorem and physical interpretation apply.

What is the difference between symmetric and self-adjoint?

A symmetric operator agrees with its adjoint on its domain, but self-adjointness additionally requires the domains to coincide; only genuinely self-adjoint operators admit the spectral theorem and generate unitary evolutions.

Unbounded Operators

Unbounded operators, such as differentiation and multiplication by an unbounded function, are not defined on the whole space; making them rigorous requires careful attention to their domains and to self-adjointness.

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Definition

An unbounded operator is a linear map defined only on a dense subspace of a Hilbert space whose norm is not bounded; the analysis centers on specifying its domain and determining whether it is self-adjoint, the condition required for a spectral decomposition.

Scope

This topic covers densely defined operators and the role of the domain, closed and closable operators and the graph, the adjoint of an unbounded operator, the distinction between symmetric and self-adjoint operators, criteria for self-adjointness and essential self-adjointness, the spectral theorem for unbounded self-adjoint operators, and Stone's theorem linking them to unitary groups.

Core questions

Why must the domain of an unbounded operator be specified so carefully?
How does the adjoint of an unbounded operator differ from the bounded case?
What separates a symmetric operator from a genuinely self-adjoint one?
How does the spectral theorem extend to unbounded self-adjoint operators?

Key theories

Spectral theorem for unbounded self-adjoint operators: Every self-adjoint operator, bounded or not, has a spectral decomposition as an integral against a projection-valued measure over its real spectrum, the result that makes such operators the rigorous model for quantum observables.
Stone's theorem on one-parameter unitary groups: Strongly continuous one-parameter groups of unitary operators correspond exactly to self-adjoint generators, identifying the self-adjoint operator behind a quantum time evolution and connecting it to dynamics.

Clinical relevance

Unbounded self-adjoint operators are the observables of quantum mechanics, including position, momentum, and the Hamiltonian; the careful theory of domains and self-adjointness determines whether a quantum system has a well-defined, unitary time evolution, making the subject indispensable to mathematical physics.

History

Von Neumann developed the rigorous theory of unbounded self-adjoint operators around 1929 to provide quantum mechanics with sound foundations, distinguishing symmetric from self-adjoint operators. Stone's theorem of 1932 tied self-adjoint generators to unitary time evolution.

Key figures

John von Neumann
Marshall Stone
Hermann Weyl

Seminal works

reedsimon1980
schmudgen2012

Frequently asked questions

Why is the domain of an unbounded operator so important?: An unbounded operator cannot act on every vector, so it is defined only on a dense subspace; the choice of that domain determines whether the operator is self-adjoint and therefore whether the spectral theorem and physical interpretation apply.
What is the difference between symmetric and self-adjoint?: A symmetric operator agrees with its adjoint on its domain, but self-adjointness additionally requires the domains to coincide; only genuinely self-adjoint operators admit the spectral theorem and generate unitary evolutions.