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Concept# Hermitian adjoint

Summary

In mathematics, specifically in operator theory, each linear operator on an inner product space defines a Hermitian adjoint (or adjoint) operator on that space according to the rule
where is the inner product on the vector space.
The adjoint may also be called the Hermitian conjugate or simply the Hermitian after Charles Hermite. It is often denoted by A† in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators are represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).
The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces . The definition has been further extended to include unbounded densely defined operators whose domain is topologically dense in - but not necessarily equal to -
Consider a linear map between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator fulfilling
where is the inner product in the Hilbert space , which is linear in the first coordinate and antilinear in the second coordinate. Note the special case where both Hilbert spaces are identical and is an operator on that Hilbert space.
When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, of an operator , where are Banach spaces with corresponding norms . Here (again not considering any technicalities), its adjoint operator is defined as with
I.e., for .
The above definition in the Hilbert space setting is really just an application of the Banach space case when one identifies a Hilbert space with its dual. Then it is only natural that we can also obtain the adjoint of an operator , where is a Hilbert space and is a Banach space. The dual is then defined as with such that
Let be Banach spaces. Suppose and , and suppose that is a (possibly unbounded) linear operator which is densely defined (i.e., is dense in ).

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