Difference between revisions of "Resolvent set"
From Encyclopedia of Mathematics
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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]]). | ||
====References==== | ====References==== | ||
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====Comments==== | ====Comments==== | ||
| − | I.e., | + | 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==== | ====References==== | ||
<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=18476
Resolvent set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Resolvent_set&oldid=18476
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article