Adjoint (operator theory): Difference between revisions

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imported>Hendra I. Nurdin
imported>Hendra I. Nurdin
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<math>\scriptstyle D(T^*)=\{ v \in H \mid \mathop{\sup}_{u \in D(T)}|\langle Tu,v\rangle_H| < \infty \}</math> defined as the map  
<math>\scriptstyle D(T^*)=\{ v \in H \mid \mathop{\sup}_{u \in D(T)}|\langle Tu,v\rangle_H| < \infty \}</math> defined as the map  
:<math>
:<math>
T^*v = v^* \quad \forall v \in D(T),  
T^*v = v^* \quad \forall v \in D(T^*),  
</math>
</math>
where for each ''v'' in ''D(T)'', ''v<sup>*</sup>'' is the unique element of ''H'' such that <math>\scriptstyle \langle u,v^* \rangle =\langle Tu,v\rangle_H</math> for all ''u'' in ''D(T)''. Additionally, if ''T'' is a bounded operator then ''T<sup>*</sup>'' is the unique bounded operator satisfying  
where for each ''v'' in ''D(T)'', ''v<sup>*</sup>'' is the unique element of ''H'' such that <math>\scriptstyle \langle u,v^* \rangle =\langle Tu,v\rangle_H</math> for all ''u'' in ''D(T)''. Additionally, if ''T'' is a bounded operator then ''T<sup>*</sup>'' is the unique bounded operator satisfying  

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In mathematics, the adjoint of an operator is a generalization, to linear operators on complex Hilbert spaces, of the notion of the Hermitian conjugate of a complex matrix.

Main idea

Consider a complex matrix M. Apart from being an array of complex numbers, M can also be viewed as a linear map or operator from 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 , is the unique linear operator on satisfying:

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 ) could be defined generally as an operator T* on H satisfying:

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 . Since T has a dense domain in H and is a continuous functional on D(T) for any , f can be extended to be a unique continuous linear functional on on H. By the Riesz representation theorem there is a unique element such that for all u in H. A linear operator with domain D(T*) = K(T) may now be defined as the map

By construction, the operator satisfies:

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 defined as the map

where for each v in D(T), v* is the unique element of H such that for all u in D(T). Additionally, if T is a bounded operator then T* is the unique bounded operator satisfying

Further reading

  1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980.
  2. K. Parthasarathy, An Introduction to Quantum Stochastic Calculus, ser. Monographs in Mathematics, Basel, Boston, Berlin: Birkhauser Verlag, 1992.