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Point Spectrum
In mathematics, particularly in functional analysis, the spectrum of a bounded linear operator (or, more generally, an unbounded linear operator) is a generalisation of the set of eigenvalues of a matrix. Specifically, a complex number \lambda is said to be in the spectrum of a bounded linear operator T if T-\lambda I * either has ''no'' set-theoretic inverse; * or the set-theoretic inverse is either unbounded or defined on a non-dense subset. Here, I is the identity operator. By the closed graph theorem, \lambda is in the spectrum if and only if the bounded operator T - \lambda I: V\to V is non-bijective on V. 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 spe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Resolvent Set
In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism. Definitions Let ''X'' be a Banach space and let L\colon D(L)\rightarrow X be a linear operator with domain D(L) \subseteq X. Let id denote the identity operator on ''X''. For any \lambda \in \mathbb, let :L_ = L - \lambda\,\mathrm. A complex number \lambda is said to be a regular value if the following three statements are true: # L_\lambda is injective, that is, the corestriction of L_\lambda to its image has an inverse R(\lambda, L)=(L-\lambda \,\mathrm)^ called the resolvent; # R(\lambda,L) is a bounded linear operator; # R(\lambda,L) is defined on a dense subspace of ''X'', that is, L_\lambda has dense range. The resolvent set of ''L'' is the set of all regular values of ''L'': :\rho(L) = \. The spectrum is the complement of the resolve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |