The adjoint operator 
of a bounded operator 
between Hilbert spaces
is the unique bounded operator satisfying
 |
(1)
|
for all 
and 
,
where the inner products are taken to be conjugate-linear
in their first arguments.
In orthonormal bases, the matrix of 
is the conjugate transpose of the matrix
of 
,
so
 |
(2)
|
For arbitrary bases having Gram matrices 
and 
, the corresponding formula is
 |
(3)
|
Thus the identification of an adjoint with a conjugate transpose depends on the choice of orthonormal bases.
For bounded operators 
and 
for which the product is defined,
 |
(4)
|
For densely defined unbounded linear operators, the domain of the adjoint must also be specified, and product formulas require additional domain hypotheses.
