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Difference between revisions of "Resolvent set"

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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816101.png" /> of complex numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816103.png" /> is a linear operator in a Banach space, for which there is an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816104.png" /> which is bounded and has a dense domain of definition in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816105.png" />. The set complementary to the resolvent set is the spectrum of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816106.png" /> (cf. [[Spectrum of an operator|Spectrum of an operator]]).
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The set $\rho(T)$ of complex numbers $z$, where $T$ is a linear operator in a Banach space, for which there is an operator $R_z=(T-zI)^{-1}$ which is bounded and has a dense domain of definition in $X$. The set complementary to the resolvent set is the spectrum of the operator $T$ (cf. [[Spectrum of an operator|Spectrum of an operator]]).
  
 
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I.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816107.png" /> is in the resolvent set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816108.png" /> if the range of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r0816109.png" /> is dense and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r08161010.png" /> has a continuous inverse. This inverse is often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r08161011.png" />, and it is called the resolvent (at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r08161012.png" />) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081610/r08161013.png" />.
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I.e., $z\in\mathbf C$ is in the resolvent set of $T$ if the range of $T-zI$ is dense and $T-zI$ has a continuous inverse. This inverse is often denoted by $R(z;T)$, and it is called the resolvent (at $z$) of $T$.
  
 
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1978)  pp. 209ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)  pp. 188, 253</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,  "Functional analysis" , Springer  (1978)  pp. 209ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Reed,  B. Simon,  "Methods of modern mathematical physics" , '''1. Functional analysis''' , Acad. Press  (1972)  pp. 188, 253</TD></TR></table>

Latest revision as of 13:48, 27 September 2014

The set $\rho(T)$ of complex numbers $z$, where $T$ is a linear operator in a Banach space, for which there is an operator $R_z=(T-zI)^{-1}$ which is bounded and has a dense domain of definition in $X$. The set complementary to the resolvent set is the spectrum of the operator $T$ (cf. Spectrum of an operator).

References

[1] F. Riesz, B. Szökevalfi-Nagy, "Leçons d'analyse fonctionelle" , Akad. Kiado (1952)


Comments

I.e., $z\in\mathbf C$ is in the resolvent set of $T$ if the range of $T-zI$ is dense and $T-zI$ has a continuous inverse. This inverse is often denoted by $R(z;T)$, and it is called the resolvent (at $z$) of $T$.

References

[a1] K. Yosida, "Functional analysis" , Springer (1978) pp. 209ff
[a2] M. Reed, B. Simon, "Methods of modern mathematical physics" , 1. Functional analysis , Acad. Press (1972) pp. 188, 253
How to Cite This Entry:
Resolvent set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolvent_set&oldid=33424
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article