Suppose is a Hilbert space with an inner product . Consider a continuous linear operator . One can show that there exists a unique continuous linear operator with the following property:
for all .
This is a generalization of the concept of an adjoint matrix (also known as a conjugate transpose, Hermitian conjugate, or Hermitian adjoint). The adjoint of an matrix with complex entries is the matrix whose entries are defined by
.
As such,
where is the transpose matrix of and is the matrix with complex conjugate entries of .
Hermitian matrices
A matrix, , is said to be Hermitian if
where is the Hermitian adjoint of .
Notation
The notation used here for the adjoint, , is commonly used in linear algebraic circles (as is ). In quantum mechanics, is exclusively used for the adjoint while is interpreted as the same thing as .