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Point Spectrum


The point spectrum of a linear operator A is the set

 sigma_p(A)={lambda in C:Ker(lambdaI-A)!={0}}.

Here I is the identity operator. Equivalently, lambda belongs to the point spectrum when lambdaI-A is not injective, so its kernel contains a nonzero vector x. This is the equation Ax=lambdax. Thus the point spectrum is exactly the set of eigenvalues of A, and it is a subset of the operator spectrum.


See also

Eigenvalue, Kernel, Operator Spectrum, Resolvent Set

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References

Conway, J. B. A Course in Functional Analysis. New York: Springer-Verlag, 1990.

Cite this as:

Weisstein, Eric W. "Point Spectrum." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PointSpectrum.html

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