Hermitian adjoint

In the functional analysis may, at any densely defined linear operator an adjoint operator (sometimes called dual operator) are defined.

Linear operators can be defined between two vector spaces with a common base ( or) adjoint operators are, however, often being considered only on Hilbert spaces, ie for example ( finite-dimensional ) Euclidean spaces. On finite-dimensional spaces corresponding to the adjoint operator of the adjoint matrix. In the Matrix with real entries, the formation of the adjoint operator corresponds to the transpose, with complex entries, the (complex) Conjugate and Transpose the output matrix. In physics and engineering sciences is therefore called in analogy to the matrix theory, the adjoint operator is usually not with but with.

Definition

In this section, the adjoint of an operator between Hilbert spaces is defined. The first sub- section is limited to so bounded continuous operators in the second section is the concept extended to unbounded operators.

Bounded operators

Let and be Hilbert spaces and a linear bounded operator. The adjoint operator is given by the equation

Defined.

Alternatively you can look for each of the picture. This is defined on the whole Hilbert space, linear continuous functional. The representation theorem of Fréchet - Riesz states that for every continuous linear functional exists a uniquely determined element, so that applies to everyone. So overall there with exactly one element for each. Now is set. This construction is equivalent to the above definition.

Unbounded operators

Let and be Hilbert spaces. With the domain of the linear operator is called. The operators and are called adjoint to each other formally, if

For all and true. Under these conditions is not given uniquely by in general. Is tightly defined, then there exists a maximum to formally adjoint operator. This is called the adjoint operator of.

Examples

  • If you choose the Hilbert space of the finite unitary vector space, then a continuous linear operator on this Hilbert space are represented by a matrix. The adjoint operator is to be represented by the corresponding adjoint matrix. Therefore, the adjoint operator is a generalization of the adjoint matrix.
  • In this example, the Hilbert space of square integrable functions is considered. With an appropriate function (for example) is the integral operator

Properties

Be densely defined. Then:

  • Is dense, it is, that is, and
  • . Here, Ker is the kernel of the operator and Ran ( for Range ) for the image space.
  • Is bounded if and only if is bounded. In which case
  • Is limited, so is the unique continuation of on

Be densely defined. The operator is defined by for. Is tightly defined, so is. Is limited, so even the equality holds.

Be a Hilbert space and. Then the sequential execution or composition is by and defined by for. Is tightly defined, the following applies. Is limited, one obtains.

Symmetric and self-adjoint operators

A linear operator is called

  • Symmetric or formally self-adjoint if for all.
  • Essentially self-adjoint, if symmetrical, densely defined and its seclusion is self-adjoint.
  • Self-adjoint if tightly defined and valid.

There is also the concept of Hermitian operator. This is mainly used in physics, but not uniformly defined.

Generalization to Banach spaces

Adjoint operators can also be defined on general Banach spaces. For a Banach space called the topological dual space. In the following, the notation is used for and. Let and be Banach spaces and be a continuous, linear operator. The adjoint operator

Is defined by

Defined for all. To distinguish these adjoint operator of the adjoint operators on Hilbert spaces, they are often quoted at a place with a.

However, if the operator is not continuous but densely defined, then we define the adjoint operator

By

The operator is always closed, it being possible. Is a reflexive Banach space and, then if and only defined tight when is lockable. Particularly applies.

Different conventions

In particular, in the linear complex case is used for the dual operator also held ( transposition, and transition to the complex conjugate ) to avoid confusion with the complex conjugate matrix. The latter is also described, but this is reserved by physicists rather for the averaging.

Domain of a function Closed operator
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