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Operator Spectrum


Let T be a linear operator on a separable Hilbert space. The spectrum sigma(T) of T is the set of lambda such that (T-lambdaI) is not invertible on all of the Hilbert space, where the lambdas are complex numbers and I is the identity operator. The definition can also be stated in terms of the resolvent of an operator

 rho(T)={lambda:(T-lambdaI) is invertible},

and then the spectrum is defined to be the complement of rho(T) in the complex plane. It is easy to demonstrate that rho(T) is an open set, which shows that the spectrum sigma(T) is closed.

If Omega is a domain in R^d (i.e., a Lebesgue measurable subset of R^d with finite nonzero Lebesgue measure), then a set Lambda subset R^d is a spectrum of Omega if {e^(2piixlambda)}_(lambda in Lambda) is an orthogonal basis of L^2(Omega) (Iosevich et al. 1999).


See also

Fuglede's Conjecture, Hilbert Space, Orthogonal Basis, Spectral Theorem, Spectrum

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References

Iosevich, A.; Katz, N. H.; and Tao, T. "Convex Bodies with a Point of Curvature Do Not Have Fourier Bases." 23 Nov 1999. http://arxiv.org/abs/math.CA/9911167.Rudin, W. Functional Analysis, 2nd ed. New York: McGraw-Hill, 1991.

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Operator Spectrum

Cite this as:

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

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