Spectrum (functional analysis) | Wikipedia audio article
Channel: wikipedia tts
Date: 06/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Spectrum_(functional_analysis)


00:03:22 1 Spectrum of a bounded operator
00:03:33 1.1 Definition
00:06:52 1.2 Spectrum and eigenvalues
00:14:29 1.3 Basic properties
00:17:27 2 Classification of points in the spectrum of an operator
00:19:44 2.1 Point spectrum
00:20:00 2.2 Approximate point spectrum
00:20:24 2.3 Residual spectrum
00:25:31 2.4 Continuous spectrum
00:26:29 2.5 Peripheral spectrum
00:27:48 2.6 Example
00:28:09 3 Further results
00:28:58 4 Spectrum of an unbounded operator
00:29:53 5 Spectrum of a unital Banach algebra
00:33:43 6 See also
00:34:28 7 References



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SUMMARY
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In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix.  Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics.
The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues.  However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues.  For example, consider the right shift operator R on the Hilbert space ℓ2,

  
    
      
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    {\displaystyle (x_{1},x_{2},\dots )\mapsto (0,x_{1},x_{2},\dots ).}
  This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc.  On the other hand, 0 is in the spectrum because the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range.  In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum.
The notion of spectrum extends to densely defined unbounded operators. In this case a complex number λ is said to be in the spectrum of such an operator T:D→X (where D is dense in X) if there is no bounded inverse (λI − T)−1:X→D.  If T is a closed operator (which includes the case that T is a bounded operator), boundedness of such inverses follows automatically if the inverse exists at all.
The space of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra.  Since the definition of the spectrum does not mention any properties of B(X) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.