Normal eigenvalue: Difference between revisions

Revision as of 08:54, 23 September 2019

Revision as of 03:43, 24 September 2019

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==Root lineal==

Let <math>\mathfrak{B}</math> be a [[Banach space]]. The [[Generalized_eigenvector#Root_lineal_of_a_linear_operator_in_a_Banach_space|root lineal]] <math>\mathfrak{L}_\lambda(A)</math> of a linear operator <math>A:\,\mathfrak{B}\to\mathfrak{B}</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_bigcup_{k\in\N}\{x\in\mathfrak{D}(A):\,(A-\lambda I_{\mathfrak{B}})^j x\in\mathfrak{D}(A)\,\forall j\in\N,\,j\le k;\, (A-\lambda I_{\mathfrak{B}})^k x=0\}\subset\mathfrak{B}</math>, where <math>I_{\mathfrak{B}}</math> is the identity operator in <math>\mathfrak{B}</math>.

where <math>I_{\mathfrak{B}}</math> is the identity operator in <math>\mathfrak{B}</math>.

This set is a [[linear manifold]] but not necessarily a [[vector space]], since it is not necessarily closed in <math>\mathfrak{B}</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>.

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The spectrum of a closed operator <math>A:\,\mathfrak{B}\to\mathfrak{B}</math> in the Banach space <math>\mathfrak{B}</math> can be decomposed into the union of two disjoint sets, the set of normal eigenvalues and the fifth type of the [[essential spectrum]]:

:<math>

\sigma(A)=\{\mathrmtext{normal\ eigenvalues\ of}\ A\}\cup\sigma_{\mathrm{ess},5}(A).

</math>