a non-trivial subspace E ⊂ H, which is itself a Hilbert space. The associated projection
operatorPEmay be constructed uniquely in terms of an orthonormal basis{|ǫp〉}{p}ofE, by
PE=
∑
p|ǫp〉〈ǫp| (3.31)
Of special interest is the projection operators onto a single vector|α〉∈H, given by
Pα=|α〉〈α| (3.32)
In quantum theory,Pαplays the role of the analyzer in the photon experiment.
• TheinverseA−^1 of an operatorAis defined as usual byAA−^1 =A−^1 A=IH.
• Theadjointof an operator is defined as follows. Given an operatorA, with domain
D(A), we define the adjoint operatorA†to be such that for all|φ〉∈D(A), we have^3
(|ψ〉,A|φ〉) = (A†|ψ〉,|φ〉) ⇔ 〈ψ|A|φ〉=〈φ|A†|ψ〉∗ (3.33)
For general operatorsA, this relation may not hold for all|ψ〉 ∈ H, but will hold only for
a dense subset ofH, which is, by definition, the domainD(A†) of the operatorA†. For a
finite-dimensional Hilbert space, the adjointA†of an operatorAis the same as the Hermitian
conjugate of that operator, and we haveA†= (A∗)t.
• Aself-adjoint operatoris an operatorAwhose adjointA†satisfiesA†=A, and whose
domain satisfies D(A†) =D(A). For a finite-dimensional Hilbert space, a self-adjoint op-
erator is simply a Hermitian operator satisfyingA†=A. For infinite dimensional Hilbert
spaces, self-adjoint is a stronger requirement than Hermitian as the domains have to coincide.
Every projection operator is self-adjoint.
• An operatorAis abounded operatorprovided that for all|φ〉∈H, we have
||A|φ〉||^2 ≤CA||φ||^2 (3.34)
Here,CAis a real positive constant which depends only on the operatorA, and not on the
vector|φ〉. Every operator in a finite-dimensional Hilbert space is automaticallya bounded
operator. In an infinite-dimensional Hilbert space, bounded operators are the closest in
properties to finite-dimensional matrices. In particular, the domain of a bounded operator
is the entire Hilbert spaceD(A) =H, and its adjointA†may be simply defined by
(|ψ〉,A|φ〉) = (A†|ψ〉,|φ〉) ⇔ 〈ψ|A|φ〉=〈φ|A†|ψ〉∗ (3.35)
(^3) The the sake of maximal clarity, we exhibit also the inner product notation here.