Hermitian adjoint

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint (or adjoint operator). Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number. In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding Hermitian adjoint (or adjoint operator). Adjoints of operators generalize conjugate transposes of square matrices to (possibly) infinite-dimensional situations. If one thinks of operators on a complex Hilbert space as generalized complex numbers, then the adjoint of an operator plays the role of the complex conjugate of a complex number. In a similar sense, one can define an adjoint operator for linear (and possibly unbounded) operators between Banach spaces. The adjoint of an operator A may also be called the Hermitian conjugate or Hermitian transpose (after Charles Hermite) of A and is denoted by A∗ or A† (the latter especially when used in conjunction with the bra–ket notation). Confusingly, A∗ may also be used to represent the conjugate of A. Consider a linear operator A : H 1 → H 2 {displaystyle A:H_{1} o H_{2}} between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator A ∗ : H 2 → H 1 {displaystyle A^{*}:H_{2} o H_{1}} fulfilling where ⟨ ⋅ , ⋅ ⟩ H i {displaystyle langle cdot ,cdot angle _{H_{i}}} is the inner product in the Hilbert space H i {displaystyle H_{i}} . Note the special case where both Hilbert spaces are identical and A {displaystyle A} is an operator on some Hilbert space. When one trades the dual pairing for the inner product, one can define the adjoint of an operator A : E → F {displaystyle A:E o F} , where E , F {displaystyle E,F} are Banach spaces with corresponding norms ‖ ⋅ ‖ E , ‖ ⋅ ‖ F {displaystyle |cdot |_{E},|cdot |_{F}} . Here (again not considering any technicalities), its adjoint operator is defined as A ∗ : F ∗ → E ∗ {displaystyle A^{*}:F^{*} o E^{*}} with I.e., ( A ∗ f ) ( u ) = f ( A u ) {displaystyle left(A^{*}f ight)(u)=f(Au)} for f ∈ F ∗ , u ∈ E {displaystyle fin F^{*},uin E} . Note that 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 A : H → E {displaystyle A:H o E} , where H {displaystyle H} is a Hilbert space and E {displaystyle E} is a Banach space. The dual is then defined as A ∗ : E ∗ → H {displaystyle A^{*}:E^{*} o H} with A ∗ f = h f {displaystyle A^{*}f=h_{f}} such that Let ( E , ‖ ⋅ ‖ E ) , ( F , ‖ ⋅ ‖ F ) {displaystyle left(E,|cdot |_{E} ight),left(F,|cdot |_{F} ight)} be Banach spaces. Suppose A : E ⊃ D ( A ) → F {displaystyle A:Esupset D(A) o F} is a (possibly unbounded) linear operator which is densely defined (i.e., D ( A ) {displaystyle D(A)} is dense in E {displaystyle E} ). Then its adjoint operator A ∗ {displaystyle A^{*}} is defined as follows. The domain is