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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.
For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on . The infinitesimal generator of this group is the system Hamiltonian .
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.
The name spectral theory was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid , in an infinite-dimensional setting.
Let be a Hilbert space and an unbounded (i.e. not necessarily bounded) linear operator with a dense domain . This condition holds automatically when H {\displaystyle H} is finite-dimensional since Dom A = H {\displaystyle \operatorname {Dom} A=H} for every linear operator on a finite-dimensional space.
Theorem — Let X be a Banach space, C be a compact operator acting on X, and σ(C) be the spectrum of C. Every nonzero λ ∈ σ(C) is an eigenvalue of C. For all nonzero λ ∈ σ(C), there exist m such that Ker((λ − C) m) = Ker((λ − C) m+1), and this subspace is finite-dimensional. The eigenvalues can only accumulate at 0.
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 ) .
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