Completed operator

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

definition

Let and be normalized spaces , a subspace and a linear operator. It is called the graph of and called him . The graph of is a subspace of the normalized space .

It is said to be closed if the graph is a closed subspace.

One calls lockable when closed subspace of the graph is a linear operator; this linear operator is then called the closure of and denoted by.

The term graph of a function or an operator is actually dispensable, because in a set-theoretical definition of the function , the function is defined by its graph. Then you can talk directly about the seclusion or the closure of .

Characterizations

  • The above designations are terminated when the following applies:
If a sequence is in with and , then is and .
This is often found as the definition of the closure of operators. It is merely a matter of characterizing the isolation of in metric space by means of sequences.
  • If and are Banach spaces, then a linear operator is closed if and only if the domain of definition is complete with the so-called graph norm defined by .
  • Furthermore, it can be concluded if and only if the following applies: If a sequence in with and converges to a , then is .

Examples

  • Let be the Banach space of continuous functions with the supremum norm , the subspace of continuously differentiable functions and be the derivative operator, i.e. H. . This operator is complete. This is apparently equivalent to a well-known theorem from elementary analysis about limit values ​​of differentiable functions, which is discussed in the article Uniform Convergence under Differentiability .
  • If the sequence space of the quadratically summable sequences is with the usual Hilbert space norm, and is defined by , then is a closed operator that is not continuous.
  • We consider the Hilbert space again . Let be the dense subspace of all finite sequences. Then the operator defined by can not be locked. (Note that the series in the above definition is always finite, i.e. it is well-defined.)
  • If continuous, then it is complete, because from and follows immediately because of the continuity . If and are Banach spaces, then the converse applies. That is precisely the statement of the famous theorem of the closed graph .

Hilbert dreams

Be and Hilbert dreams and as above. It is said to be dense if the subspace is dense. In this case the adjoint operator of is explained. This simplifies the investigation of closable or closed operators, because the following statements apply to a densely-defined operator :

  • is lockable if and only if is tightly defined.
  • If it is lockable, then applies and
  • If closed, then is a self-adjoint operator .

Applications

In quantum mechanics, the proof of the self adjointness of densely-defined operators in Hilbert spaces is of fundamental importance, because such operators are precisely the quantum mechanical observables . It is often quite easy to prove that the operator in question is symmetric . Then the following sentence can help:

Let be a Hilbert space, a dense subspace and a closed and symmetric operator. Then the following statements are equivalent, where be the identical operator.

  • is self adjoint.
  • The operators are injective .
  • The operators are surjective .
  • The operators have dense picture in .

Here, i is the imaginary unit, and the definition section , and is of or .

In quantum mechanics, one often does not consider the self-adjoint operators on their complete domain of definition, but only on a subspace, the elements of which have pleasant properties. Operators defined in spaces are therefore often restricted to spaces of differentiable functions, e.g. B. on spaces of arbitrarily often differentiable functions, especially if the considered operators are differential operators . In doing so, one chooses such subspaces so that the termination of the restricted operator is again . Such subspaces are called an essential area or core of , which must not be confused with the null space , which is also called the core. Many quantum mechanical calculations are only carried out on such kernels, after which the relationships found between operators are continued with the closing operation.

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