Adjoint of an operator

In mathematics, specifically in functional analysis, each bounded linear operator on a complex Hilbert space has a corresponding 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 there can be defined an adjoint operator for linear (and possibly unbounded) operators between Banach spaces.

The adjoint of an operator A may also be called the Hermitian adjoint, 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).

Consider a linear operator ${\displaystyle A:H_{1}\to H_{2}}$ between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator ${\displaystyle A^{*}:H_{2}\to H_{1}}$ fulfilling

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