If A*A is a compact operator, is A compact?

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SUMMARY

If A is a continuous and linear operator and A*A is compact, then A must also be compact. This conclusion is derived from the properties of compact operators in functional analysis. The polar decomposition can be utilized to demonstrate this relationship, as it provides a framework for understanding the structure of operators and their adjoints.

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  • Understanding of linear operators in functional analysis
  • Familiarity with compact operators and their properties
  • Knowledge of adjoint operators (A*)
  • Basic concepts of polar decomposition
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Mathematicians, graduate students in functional analysis, and anyone studying operator theory will benefit from this discussion.

elina
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Hi, if I know that A is continuous and linear and A*A is compact, where A* is the adjungate, how will I show that A has to be compact?
 
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You could try using the http://en.wikipedia.org/wiki/Polar_decomposition" .
 
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