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Let and be Hilbert spaces, and let : be an unbounded operator from into . Suppose that is a closed operator and that is densely defined, that is, is dense in . Let : denote the adjoint of .
A Hilbert space is a vector space equipped with an inner product operation, which allows lengths and angles to be defined. Furthermore, Hilbert spaces are complete, which means that there are enough limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space.
Lemma — If A, B are bounded operators on a Hilbert space H, and A*A ≤ B*B, then there exists a contraction C such that A = CB. Furthermore, C is unique if Ker ( B* ) ⊂ Ker ( C ). The operator C can be defined by C ( Bh ) = Ah , extended by continuity to the closure of Ran ( B ), and by zero on the orthogonal complement of Ran( B ) .
Normed space – Vector space on which a distance is defined; Operator algebra – Branch of functional analysis; Operator theory – Mathematical field of study; Topologies on the set of operators on a Hilbert space; Unbounded operator – Linear operator defined on a dense linear subspace
In operator theory, an area of mathematics, Douglas' lemma [1] relates factorization, range inclusion, and majorization of Hilbert space operators. It is generally attributed to Ronald G. Douglas, although Douglas acknowledges that aspects of the result may already have been known. The statement of the result is as follows:
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operator N : H → H that commutes with its Hermitian adjoint N*, that is: NN* = N*N. [1] Normal operators are important because the spectral theorem holds for them. The class of normal operators is well understood.
A rigged Hilbert space is a pair (H, Φ) with H a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure for which the inclusion map:, is continuous. [ 4 ] [ 5 ] Identifying H with its dual space H * , the adjoint to i is the map i ∗ : H = H ∗ → Φ ∗ . {\displaystyle i^{*}:H=H^{*}\to \Phi ^{*}.}
The space is a fixed complex Hilbert space of countably infinite dimension. The observables of a quantum system are defined to be the (possibly unbounded ) self-adjoint operators A {\displaystyle A} on H {\displaystyle \mathbb {H} } .