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Línea 102:
==Espectro de operadores no acotados==
La definición de espectro
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== Spectrum of unbounded operators ==▼
One can extend the definition of spectrum for [[unbounded operator]]s on a [[Banach space]] ''X'', operators which are no longer elements in the Banach algebra ''B''(''X''). One proceeds in a manner similar to the bounded case. A complex number <math>\lambda</math> is said to be in the '''resolvent set''', that is, the [[complement (set theory)|complement]] of the spectrum of a linear operator▼
:<math>T: D \subset X \to X</math>▼
if the operator ▼
:<math>T-\lambda I: D \to X</math> ▼
has a bounded inverse, i.e. if there exists a bounded operator ▼
:<math>S : X \rightarrow D</math>▼
such that▼
:<math>S (T - \lambda) = I_D, \, (T - \lambda) S = I_X.</math> ▼
A complex number <math>\lambda</math> is then in the '''spectrum''' if this property fails to hold. One can classify the spectrum in exactly the same way as in the bounded case. ▼
The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.▼
Immediately from the definition, it can be deduced that ''S'' can not be invertible, in the sense of bounded operators. Since the domain ''D'' may be a proper subset of ''X'', the expression ▼
:<math>\, (T - \lambda) S = I_X</math>▼
makes sense only if ''Ran''(''S'') is contained in ''D''. Similarly, ▼
:<math>\, S (T - \lambda) = I_D</math> ▼
implies ''D'' ⊂ ''Ran''(''S''). Therefore, ''λ'' being in the resolvent set of ''T'' means ▼
:<math>T-\lambda I: D \to X</math>▼
is bijective. (Recall that bijectivity of ''T - λ'' is not implied by invertibility if ''T'' is bounded.)▼
The converse is true if one introduces the additional assumption that ''T'' is closed. By the [[closed graph theorem]], if ''T - λ'': ''D'' → ''X'' is bijective, then its (algebraic) inverse map is necessarily a bounded operator. (Notice the completeness of ''X'' is required in invoking the closed graph theorem.) Therefore, in contrast to the bounded case, the condition that a complex number ''λ'' lie in the spectrum of ''T'' becomes a purely algebraic one: for a closed <math>T</math>, <math>\lambda</math> is in the spectrum of <math>T</math> if and only if <math>T-\lambda</math> is not bijective.▼
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==Ejemplos==
===Operador momento lineal===
Línea 151 ⟶ 191:
*Richtmyer, Robert D. (1978): ''Principles of advanced mathematical physics'', Springer-Verlag, New York, ISBN 0-387-08873-3.
▲<!--
▲== Spectrum of unbounded operators ==
▲One can extend the definition of spectrum for [[unbounded operator]]s on a [[Banach space]] ''X'', operators which are no longer elements in the Banach algebra ''B''(''X''). One proceeds in a manner similar to the bounded case. A complex number <math>\lambda</math> is said to be in the '''resolvent set''', that is, the [[complement (set theory)|complement]] of the spectrum of a linear operator
▲:<math>T: D \subset X \to X</math>
▲if the operator
▲:<math>T-\lambda I: D \to X</math>
▲has a bounded inverse, i.e. if there exists a bounded operator
▲:<math>S : X \rightarrow D</math>
▲such that
▲:<math>S (T - \lambda) = I_D, \, (T - \lambda) S = I_X.</math>
▲A complex number <math>\lambda</math> is then in the '''spectrum''' if this property fails to hold. One can classify the spectrum in exactly the same way as in the bounded case.
▲The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.
▲Immediately from the definition, it can be deduced that ''S'' can not be invertible, in the sense of bounded operators. Since the domain ''D'' may be a proper subset of ''X'', the expression
▲:<math>\, (T - \lambda) S = I_X</math>
▲makes sense only if ''Ran''(''S'') is contained in ''D''. Similarly,
▲:<math>\, S (T - \lambda) = I_D</math>
▲implies ''D'' ⊂ ''Ran''(''S''). Therefore, ''λ'' being in the resolvent set of ''T'' means
▲:<math>T-\lambda I: D \to X</math>
▲is bijective. (Recall that bijectivity of ''T - λ'' is not implied by invertibility if ''T'' is bounded.)
▲The converse is true if one introduces the additional assumption that ''T'' is closed. By the [[closed graph theorem]], if ''T - λ'': ''D'' → ''X'' is bijective, then its (algebraic) inverse map is necessarily a bounded operator. (Notice the completeness of ''X'' is required in invoking the closed graph theorem.) Therefore, in contrast to the bounded case, the condition that a complex number ''λ'' lie in the spectrum of ''T'' becomes a purely algebraic one: for a closed <math>T</math>, <math>\lambda</math> is in the spectrum of <math>T</math> if and only if <math>T-\lambda</math> is not bijective.
▲-->
==Referencias==
{{reflist}}
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