• Source: Resolvent set

In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.




Definitions


Let X be a Banach space and let



L
:
D
(
L
)
→
X


{\displaystyle L\colon D(L)\rightarrow X}

be a linear operator with domain



D
(
L
)
⊆
X


{\displaystyle D(L)\subseteq X}

. Let id denote the identity operator on X. For any



λ
∈

C



{\displaystyle \lambda \in \mathbb {C} }

, let





L

λ


=
L
−
λ


i
d

.


{\displaystyle L_{\lambda }=L-\lambda \,\mathrm {id} .}


A complex number



λ


{\displaystyle \lambda }

is said to be a regular value if the following three statements are true:





L

λ




{\displaystyle L_{\lambda }}

is injective, that is, the corestriction of




L

λ




{\displaystyle L_{\lambda }}

to its image has an inverse



R
(
λ
,
L
)
=
(
L
−
λ


i
d


)

−
1




{\displaystyle R(\lambda ,L)=(L-\lambda \,\mathrm {id} )^{-1}}

called the resolvent;




R
(
λ
,
L
)


{\displaystyle R(\lambda ,L)}

is a bounded linear operator;




R
(
λ
,
L
)


{\displaystyle R(\lambda ,L)}

is defined on a dense subspace of X, that is,




L

λ




{\displaystyle L_{\lambda }}

has dense range.
The resolvent set of L is the set of all regular values of L:




ρ
(
L
)
=
{
λ
∈

C

∣
λ


is a regular value of


L
}
.


{\displaystyle \rho (L)=\{\lambda \in \mathbb {C} \mid \lambda {\mbox{ is a regular value of }}L\}.}


The spectrum is the complement of the resolvent set




σ
(
L
)
=

C

∖
ρ
(
L
)
,


{\displaystyle \sigma (L)=\mathbb {C} \setminus \rho (L),}


and subject to a mutually singular spectral decomposition into the point spectrum (when condition 1 fails), the continuous spectrum (when condition 2 fails) and the residual spectrum (when condition 3 fails).
If



L


{\displaystyle L}

is a closed operator, then so is each




L

λ




{\displaystyle L_{\lambda }}

, and condition 3 may be replaced by requiring that




L

λ




{\displaystyle L_{\lambda }}

be surjective.




Properties


The resolvent set



ρ
(
L
)
⊆

C



{\displaystyle \rho (L)\subseteq \mathbb {C} }

of a bounded linear operator L is an open set.
More generally, the resolvent set of a densely defined closed unbounded operator is an open set.




Notes






References


Reed, M.; Simon, B. (1980). Methods of Modern Mathematical Physics: Vol 1: Functional analysis. Academic Press. ISBN 978-0-12-585050-6.
Renardy, Michael; Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second ed.). New York: Springer-Verlag. xiv+434. ISBN 0-387-00444-0. MR2028503 (See section 8.3)




External links


Voitsekhovskii, M.I. (2001) [1994], "Resolvent set", Encyclopedia of Mathematics, EMS Press




See also


Resolvent formalism
Spectrum (functional analysis)
Decomposition of spectrum (functional analysis)

Kata Kunci Pencarian:

• Ruang Hilbert
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• Resolvent
• Resolvent formalism
• Spectrum (functional analysis)
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• Decomposition of spectrum (functional analysis)