- Resolvent set
In
linear algebra andoperator theory , the resolvent set of alinear operator is a set ofcomplex number s for which the operator is in some sense "well-behaved ". The resolvent set plays an important role in theresolvent formalism .Definitions
Let "X" be a
Banach space and let Lcolon D(L) ightarrow X be a linear operator with domain D(L) subseteq X. Let id denote the identity operator on "X". For any lambda in mathbb{C}, let:L_{lambda} = L - lambda mathrm{id}.
lambda is said to be a regular value if R(lambda, L), the inverse operator to L_lambda
# exists;
# is abounded linear operator ;
# is defined on a dense subspace of "X".The resolvent set of "L" is the set of all regular values of "L"::ho (L) = { lambda in mathbf{C} | lambda mbox{ is a regular value of } L }.
The spectrum is the complement of the resolvent set:
:sigma (L) = mathbf{C} setminus ho (L).
The spectrum can be further decomposed into the point/discrete spectrum (where condition 1 fails), the continuous spectrum (where conditions 1 and 3 hold but condition 2 fails) and the residual/compression spectrum (where condition 1 holds but condition 3 fails).
Properties
* The resolvent set ho(L) subseteq mathbb{C} of a bounded linear operator "L" is an
open set .References
* cite book
last = Renardy
first = Michael
coauthors = Rogers, Robert C.
title = An introduction to partial differential equations
series = Texts in Applied Mathematics 13
edition = Second edition
publisher = Springer-Verlag
address = New York
year = 2004
pages = pp. xiv+434
isbn = 0-387-00444-0 MathSciNet|id=2028503 (See section 7.3)External links
* springer
id = R/r081610
title = Resolvent set
last = Voitsekhovskii
first = M.I.
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