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Essential SpectrumIn mathematics, the essential spectrum of a bounded operator is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible". The essential spectrum of self-adjoint operators In formal terms, let X be a Hilbert space and let T be a bounded self-adjoint operator on X. Definition The essential spectrum of T, usually denoted σess(T), is the set of all complex numbers λ such that - λI − T
is not a Fredholm operator. Here, an operator is Fredholm if its range is closed and its kernel and cokernel are finite-dimensional. Furthermore, I denotes the identity operator on X, so that - I(x) = x
for all x in X. Properties The essential spectrum is always closed, and it is a subset of the spectrum. Since T is self-adjoint, the spectrum is contained on the real axis. The essential spectrum is invariant under compact perturbations. That is, if K is a compact operator on X, then the essential spectra of T and that of T + K coincide. This explains why it is called the essential spectrum: Weyl (1910) originally defined the essential spectrum of a certain differential operator to be the spectrum independent of boundary conditions. Weyl's criterion for the essential spectrum is as follows. First, a number λ is in the spectrum of T if and only if there exists a sequence {ψk} in the space X such that ||ψk|| = 1 and -
Furthermore, λ is in the essential spectrum if there is a sequence satisfying this condition, but such that it contains no convergent subsequence; such a sequence is called a singular sequence. The discrete spectrum The essential spectrum is a subset of the spectrum σ, and its complement is called the discrete spectrum, so -
A number λ is in the discrete spectrum if it is an isolated eigenvalue of finite multiplicity, meaning that the dimension of the space -
is finite but non-zero and that there is an ε > 0 such that μ ∈ σ(T) and |μ−λ| < ε imply that μ and λ are equal. The essential spectrum of general bounded operators In the general case, X denotes a Banach space and T is a bounded operator on X. There are several definitions of the essential spectrum in the literature, which are not equivalent. - The essential spectrum σess,1(T) is the set of all λ such that λI − T is not semi-Fredholm (an operator is semi-Fredholm if its range is closed and its kernel or its cokernel is finite-dimensional).
- The essential spectrum σess,2(T) is the set of all λ such that the range of λI − T is not closed or the kernel of λI − T is infinite-dimensional.
- The essential spectrum σess,3(T) is the set of all λ such that λI − T is not Fredholm (an operator is Fredholm if its range is closed and both its kernel and its cokernel are finite-dimensional).
- The essential spectrum σess,4(T) is the set of all λ such that λI − T is not Fredholm with index zero (the index of a Fredholm operator is the difference between the dimension of the kernel and the dimension of the cokernel).
- The essential spectrum σess,5(T) is the union of σess,1(T) with all components of C \ σess,1(T) that do not intersect with the resolvent set C \ σ(T).
The essential spectrum of an operator is closed, whatever definition is used. Furthermore, -
but any of these inclusions may be strict. However, for self-adjoint operators, all the above definitions for the essential spectrum coincide. Define the radius of the essential spectrum by -
Even though the spectra may be different, the radius is the same for all k. The essential spectrum σess,k(T) is invariant under compact perturbations for k = 1,2,3,4, but not for k = 5. The case k = 4 gives the part of the spectrum that is independent of compact perturbations, that is, -
where K(X) denotes the set of compact operators on X. The second definition generalizes Weyl's criterion: σess,2(T) is the set of all λ for which there is no singular sequence. References The self-adjoint case is discussed in - Michael Reed and Barry Simon (1980), Functional Analysis, Academic Press, San Diego. ISBN 0-12-585050-5.
A discussion of the spectrum for general operators can be found in - D.E. Edmunds and W.D. Evans (1987), Spectral theory and differential operators, Oxford University Press. ISBN 0-19-853542-2.
The original definition of the essential spectrum goes back to - H. Weyl (1910), Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Mathematische Annalen 68, 220–269.
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