The spectrum of a linear operator consists of the scalars for which is not invertible. In finite dimensions it is the set of eigenvalues.
An approximate eigenvalue of a bounded operator is a scalar admitting unit vectors with . Every approximate eigenvalue belongs to the spectrum; for a normal operator, all spectral points are approximate eigenvalues.
The unit vectors realizing an approximate eigenvalue are approximate eigenvectors: their residual tends to zero, although they need not converge to any genuine eigenvector.
In the disjoint convention for a bounded linear operator , the residual spectrum consists of for which is injective but has nondense range. Together with the point spectrum and continuous spectrum it partitions the spectrum of a bounded operator. Some conventions also include noninjective operators with nondense range, so the convention must be stated.
For a self-adjoint operator , a real number lies in the continuous spectrum when is injective with dense range but is not surjective. Such spectral values need not be eigenvalues of normalizable states. The free-particle Hamiltonian on the real line has a continuous energy spectrum.
For a bounded operator on a complex Banach space, the spectrum is a nonempty compact subset of the complex plane contained in the closed disk of radius equal to the operator norm.
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The term "spectrum" can refer to various concepts depending on the context in which it is used. Here are some common interpretations: 1. **Physics and Light**: In physics, a spectrum refers to the range of different wavelengths of electromagnetic radiation. This includes visible light, which can be split into a spectrum of colors (red, orange, yellow, green, blue, indigo, violet) through a prism.