adjoint
Let be a Hilbert space and let be a densely defined linear operator. Suppose that for some , there exists such that for all . Then such is unique, for if is another element of satisfying that condition, we have for all , which implies since is dense (http://planetmath.org/Dense). Hence we may define a new operator by
It is easy to see that is linear, and it is called the adjoint of .
Remark. The requirement for to be densely defined is essential, for otherwise we cannot guarantee to be well defined.
Title | adjoint |
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Canonical name | Adjoint |
Date of creation | 2013-03-22 13:48:09 |
Last modified on | 2013-03-22 13:48:09 |
Owner | Koro (127) |
Last modified by | Koro (127) |
Numerical id | 10 |
Author | Koro (127) |
Entry type | Definition |
Classification | msc 47A05 |
Synonym | adjoint operator |
Related topic | TransposeOperator |