Spectrum and Resolvent
The eigenvalues of an n × n matrix M constitute a (finite) point set in the complex plane, called the spectrum of M. If A is any linear operator in a Hilbert space ℌ, the complex plane ℂ is similarly decomposed into two parts: the spectrum of A, denoted by σ(A), and the resolvent set, denoted by ρ(A). The spectrum of A is further decomposed into the point spectrum Pσ(A), the continuous spectrum Cσ(A), and the residual spectrum Rσ(A).
KeywordsContinuous, point, and residual spectrum eigenvectors and approximate eigenvectors resolvent analyticity of the resolvent the Cayley transform von Neumann’s theory of the extension of symmetric operators the deficiency indices of a symmetric operator second definition of self-adjoint operator
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