Closed operator

Completed operators in functional analysis, a branch of mathematics is considered. These are linear operators with a particular topological feature, which is weaker than continuity. These play a significant role in the important for quantum mechanics theory of densely - defined operators.

Definition

Let and be normed spaces, a subspace and a linear operator. This is called the graph of and is denoted by. The graph of is a subspace of the normed space.

Is called complete if the graph is a closed subspace.

It's called being locked where the closed subspace of the graph of a linear operator is; this linear operator of financial statements are then called and designated.

The concept of the graph of a function or an operator is actually unnecessary, for, in a set-theoretic definition of the function, the function is defined by its graph. Then you can talk directly from the seclusion or to the conclusion of.

Characterizations

  • With above notation is just finished when the following applies:
  • Are Banach spaces and, as a linear operator if and only complete if the domain is completely defined by the so-called graph norm.
  • Further, if and lockable, if the following applies: If a sequence in with and converges to a, then.

Examples

  • Be the Banach space of continuous functions with the supremum norm, the subspace of continuously differentiable functions and be the derivative operator, ie. This operator is complete. This is obviously equivalent to a well-known theorem from elementary analysis on limit values ​​of differentiable functions, which is discussed in the article Uniform convergence under differentiability.
  • Is the result space of square summable sequences with the usual standard Hilbert space, and is defined by, as is a closed operator, which is not continuous.
  • We again consider the Hilbert space. Be the dense subspace of all finite sequences. Then the operator defined by is not lockable. ( Note that the number in the above definition is always finite, so is well-defined. )
  • Is continuous, so is complete, because of the continuity and follows immediately. Are and Banach spaces, the converse is also true. This is just the statement of the famous phrase from the closed graph.

Hilbert spaces

Let and be Hilbert spaces and above. One says, is densely defined if the subspace is dense. In this case, the adjoint operator of is explained. This simplifies the examination lockable or enclosed operators, because the following statements apply for a densely defined operator:

  • If and only lockable if it is densely defined.
  • Is lockable, applies and so
  • If completed, it is a self-adjoint operator.

Applications

In quantum mechanics, is to demonstrate the self-adjoint densely defined operator in Hilbert spaces of fundamental importance, because such operators are exactly the quantum mechanical observables. Often is the proof that the operator in question is symmetrical, pretty easy. Then, the following sentence can help you:

Be a Hilbert space, a dense subspace and a closed and symmetric operator. Then the following statements are equivalent, the identity operator is shown.

  • Is self-adjoint.
  • The operators are injective.
  • The operators are surjective.
  • The operators have dense image in.

Here i is the imaginary unit, and the domain of, or is the from and.

In quantum mechanics, one often does not consider the self-adjoint operators on its whole domain, but only on a subspace whose elements have nice properties. So you limit defined in spaces like operators on spaces of differentiable functions, eg to spaces as often differentiable functions, especially when the considered operators are differential operators. It selects one such subspaces, so that the conclusion of the restricted operator is again. Such subspaces is called a substantial portion or core of what must not be the null space, which can also be called the core, confused. Many quantum mechanical calculations are performed only on such cores, then sets we found the relations between operators as required by the final operation.

Swell

  • R.V. Kadison, JR Ringrose: Fundamentals of the Theory of Operator Algebras, 1983, ISBN 0-12-393301-3
  • H. Triebel: Higher Analysis, Verlag Harri German, ISBN 3-87144-583-5
  • Functional Analysis
Closed set Uniform norm Sequence space Closed graph theorem Hermitian adjoint Differentialoperator Richard Kadison
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