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Adjoint (operator theory)

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Main idea

Consider a complex $\scriptstyle n \times n$ matrix M. Apart from being an array of complex numbers, M can also be viewed as a linear map or operator from $\scriptstyle\mathbb{C}^n$ to itself. In order to generalize the idea of the Hermitian conjugate of a complex matrix to linear operators on more general complex Hilbert spaces, it is necessary to be able to characterize the Hermitian conjugate as an operator. The crucial observation here is the following: for any complex matrix M, its Hermitian tranpose, denoted by $\scriptstyle M^*$ , is the unique linear operator on $\scriptstyle\mathbb{C}^n$ satisfying:

$\langle Mx,y\rangle=\langle x,M^* y\rangle \quad \forall x,y \in \mathbb{C}^n.$

This suggests that the "Hermitian conjugate" or, as it is more commonly known, the adjoint of a linear operator T on an arbitrary complex Hilbert space H (with inner product $\scriptstyle \langle \cdot,\cdot \rangle_H$ ) could be defined generally as an operator T^* on H satisfying:

$\langle Tx,y\rangle_H=\langle x,T^* y\rangle_H \quad \forall x,y \in H. \quad (1)$

It turns out that this idea is almost correct. It is correct and T^* exists and is unique if T is a bounded operator on H, but additional care has to be taken on infinite dimensional Hilbert spaces since operators on such spaces can be unbounded and there may not exist an operator T^* satisfying (1).

Existence of the adjoint

Suppose that T is a densely defined operator on H with domain D(T). Consider the vector space $\scriptstyle K(T)=\{ v \in H \mid \mathop{\sup}_{u \in D(T)}|\langle Tu,v\rangle_H| < \infty \}$ . Since T has a dense domain in H and $\scriptstyle f_v(u)=\langle Tu,v\rangle_H$ is a continuous linear functional on D(T) for any $\scriptstyle v \in K(T)$ , f_v can be extended to a unique continuous linear functional $\scriptstyle \tilde f_v$ on H. By the Riesz representation theorem there is a unique element $\scriptstyle v^* \in H$ such that $\scriptstyle \tilde f_v(u)=\langle u,v^*\rangle_H$ for all u in H. A linear operator $T *$ with domain D(T^*) = K(T) may now be defined as the map

$T^*v = v^* \quad \forall v \in D(T).$

By construction, the operator $\scriptstyle T^*$ satisfies:

$\langle Tx,y\rangle_H=\langle x,T^* y\rangle_H \quad \forall x \in D(T),\,\forall y \in D(T^*). \quad (2)$

When T is a bounded operator (hence D(T) = H) then it can be shown, again using the Riesz representation theorem, that T^* is the unique bounded linear operator satisfying (2).

Formal definition of the adjoint of an operator

Let T be an operator on a Hilbert space H with dense domain D(T). Then the adjoint T^* of T is an operator with domain $\scriptstyle D(T^*)=\{ v \in H \mid \mathop{\sup}_{u \in D(T)}|\langle Tu,v\rangle_H| < \infty \}$ defined as the map

$T^*v = v^* \quad \forall v \in D(T^*),$

where for each v in D(T^*), v^* is the unique element of H such that $\scriptstyle \langle u,v^* \rangle =\langle Tu,v\rangle_H$ for all u in D(T). Additionally, if T is a bounded operator then T^* is the unique bounded operator satisfying