Fredholm index
Let P be a Fredholm operator. The index of P is defined as
index(P) | = | dimker(P)-dimcoker(P) | ||
= | dimker(P)-dimker(P*). |
Note: this is well defined as ker(P) and ker(P*) are finite-dimensional
vector spaces, for P Fredholm.
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index(P*)=-index(P).
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index(P+K)=index(P) for any compact operator
K.
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If P1:ℋ1→ℋ2 and P2:ℋ2→ℋ3 are Fredholm operators, then index(P2P1)=index(P1)+index(P2).
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If t→Pt, t∈[0,1] is a norm continuous
path of Fredholm operators, then index(Pt)=index(P0).
Fredholm operators of the form 𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒+𝑐𝑜𝑚𝑝𝑎𝑐𝑡 have index zero.
Title | Fredholm index |
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Canonical name | FredholmIndex |
Date of creation | 2013-03-22 13:20:45 |
Last modified on | 2013-03-22 13:20:45 |
Owner | mhale (572) |
Last modified by | mhale (572) |
Numerical id | 9 |
Author | mhale (572) |
Entry type | Definition |
Classification | msc 47A53 |
Synonym | index |
Related topic | FredholmOperator |