Normal eigenvalue: Difference between revisions

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In [[Spectral theory]], for [[Unbounded_operator#Closed_linear_operators|closed linear operators]] which are not necessarily [[Self-adjoint operators|self-adjoint]], the set of ''normal eigenvalues'' is defined as a subset of the [[point spectrum]] <math>\sigma_p(A)</math> of <math>A</math> such that the space <math>\mathbf{X}</math> admits a decomposition into a direct sum of a finite-dimensional [[generalized eigenspace]] and an [[invariant subspace]] where <math>A-\lambda I_{\mathbf{X}}I</math> has a bounded inverse.

==Root lineal==

Let <math>\mathbf{X}</math> be a [[Banach space]]. We recall that the [[root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathbf{X}\to\mathbf{X}</math> with ___domain <math>\mathfrak{D}(A)</math> corresponding to the eigenvalue <math>\lambda\in\sigma_p(A)</math> is defined as

:<math>\mathfrak{L}_\lambda(A)=\cup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathbf{X}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathbf{X}})^k x=0\}\subset\mathbf{X}</math>.

This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathbf{X}</math>. If this set is closed (for example, when it is finite-dimensional), it is called the [[generalized eigenspace]] of <math>A</math> corresponding to the eigenvalue <math>\lambda</math>.