Help me tie up loose ends in operator theory

In summary: This shows that D(A) is complete and hence a Banach space.In summary, the conversation is about proving that D(A) with the graph norm is a Banach space, as well as the equivalence of the two known definitions of a closed operator in a Banach space. The closedness of A is important in proving completeness of D(A) with the graph norm, and the fact that closed subsets of complete metric spaces are themselves complete can be used to show this. The equivalence of the two definitions is considered trivial by many.
  • #1
dextercioby
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Dear knowledgeable person,

please, help me with the following problems:

Let [itex] A [/itex] be a closed operator in the Banach space [itex] (B, ||.||) [/itex].
Let D(A) be its domain in B. Prove that D(A) endowed with the graph norm is a Banach space.

The graph norm is defined as:

let 'a' be a vector in D(A). Then [itex] ||a||_{\mathbb{graph}} := ||a||+ ||A a|| [/itex].

I can easily show that the graph norm makes D(A) a pre-Banach space, but what about completeness wrt the graph norm ? I'm sure it has to do with the closedness of A, but how ?

Also, how does one prove the quivalence of the 2 known definitions of a closed operator in a banach space ? Is it really trivial, as everyone claims ?

See also here
http://planetmath.org/?method=l2h&from=objects&name=ClosedOperator&op=getobj
 
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  • #2
Are you aware that closed subsets of complete metric spaces are themselves complete? That should give you what you want.
 
  • #3
It seems trivial. If (a_n) is a Cauchy sequence w.r.t. the graph norm, then both (a_n) and (Aa_n) is a Cauchy sequence w.r.t. the usual norm. Hence they converge to some a and b, respectively. As A is closed, we get that a is in D(A) and b=Aa.
 

1. What is operator theory?

Operator theory is a branch of mathematics that deals with the study of linear operators on vector spaces. It is a fundamental tool in functional analysis and has applications in many areas of mathematics and physics.

2. Why is it important to tie up loose ends in operator theory?

Tying up loose ends in operator theory is important because it ensures that the theory is consistent and complete. It allows for a deeper understanding of the subject and can lead to new discoveries and applications.

3. What are some common loose ends in operator theory?

Some common loose ends in operator theory include open questions, incomplete proofs, and unresolved conjectures. These can arise from the complexity of the subject or from the limitations of current mathematical techniques.

4. How do scientists go about tying up loose ends in operator theory?

Scientists use a variety of methods to tie up loose ends in operator theory, such as developing new mathematical techniques, finding counterexamples to conjectures, and collaborating with other researchers in the field. They also rely on rigorous mathematical proofs and experiments to test their theories.

5. What are some current areas of research in operator theory?

Some current areas of research in operator theory include non-commutative geometry, operator algebras, and spectral theory. Other topics of interest include the study of unbounded operators, C*-algebras, and applications in quantum mechanics and signal processing.

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