Hermitian adjoint

It is often denoted by A† in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics.In finite dimensions where operators can be represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spacesThe definition has been further extended to include unbounded densely defined operators, whose domain is topologically dense in, but not necessarily equal to,Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operatorNote the special case where both Hilbert spaces are identical andWhen one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, of an operatorHere (again not considering any technicalities), its adjoint operator is defined asThe above definition in the Hilbert space setting is really just an application of the Banach space case when one identifies a Hilbert space with its dual (via the Riesz representation theorem).Then it is only natural that we can also obtain the adjoint of an operatoris a (possibly unbounded) linear operator which is densely defined (i.e.,but the extension only worked for specific elementsas The fundamental defining identity is thus Suppose H is a complex Hilbert space, with inner productThen the adjoint of A is the continuous linear operator A∗ : H → H satisfying Existence and uniqueness of this operator follows from the Riesz representation theorem.[2] This can be seen as a generalization of the adjoint matrix of a square matrix which has a similar property involving the standard complex inner product.The following properties of the Hermitian adjoint of bounded operators are immediate:[2] If we define the operator norm of A by then Moreover, One says that a norm that satisfies this condition behaves like a "largest value", extrapolating from the case of self-adjoint operators.The set of bounded linear operators on a complex Hilbert space H together with the adjoint operation and the operator norm form the prototype of a C*-algebra.A densely defined operator A from a complex Hilbert space H to itself is a linear operator whose domain D(A) is a dense linear subspace of H and whose values lie in H.[3] By definition, the domain D(A∗) of its adjoint A∗ is the set of all y ∈ H for which there is a z ∈ H satisfying Owing to the density ofhold with appropriate clauses about domains and codomains.[clarification needed] For instance, the last property now states that (AB)∗ is an extension of B∗A∗ if A, B and AB are densely defined operators.is the orthogonal complement of a subspace, and therefore is closed.which, in turn, is proven through the following chain of equivalencies: The closureAs above, the word "function" may be replaced with "operator".Select a measurable, bounded, non-identically zero functionThe definition of adjoint operator requires thatA bounded operator A : H → H is called Hermitian or self-adjoint if which is equivalent to In some sense, these operators play the role of the real numbers (being equal to their own "complex conjugate") and form a real vector space.They serve as the model of real-valued observables in quantum mechanics.See the article on self-adjoint operators for a full treatment.For a conjugate-linear operator the definition of adjoint needs to be adjusted in order to compensate for the complex conjugation.An adjoint operator of the conjugate-linear operator A on a complex Hilbert space H is an conjugate-linear operator A∗ : H → H with the property: The equation is formally similar to the defining properties of pairs of adjoint functors in category theory, and this is where adjoint functors got their name from.
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