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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 T ∗ : dom ( T ∗ ) ⊆ H → G {\displaystyle T^{*}:\operatorname {dom} \left(T^{*}\right)\subseteq H\to G} denote the adjoint of T . {\displaystyle T.}
Formally, a Hilbert space is a vector space equipped with an inner product that induces a distance function for which the space is a complete metric space. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz.
In the Hilbert space formulation of quantum mechanics as presented by von Neumann, a physical observable is represented by some (possibly unbounded) densely defined self-adjoint operator A on a Hilbert space H. A has a spectral decomposition, which is a projection-valued measure E defined on the Borel subsets of R.
In the simplest case, of a POVM with a finite number of elements acting on a finite-dimensional Hilbert space, Naimark's theorem says that if {} = is a POVM acting on a Hilbert space of dimension , then there exists a PVM {} = acting on a Hilbert space ′ of dimension ′ and an isometry: ′ such that for all ,
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.
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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 of bounded linear operators B(X) on a Banach space X is an example of a unital Banach algebra. Since the definition of the spectrum does not mention any properties of B ( X ) except those that any such algebra has, the notion of a spectrum may be generalised to this context by using the same definition verbatim.