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Resolvent set

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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).

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://www.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