Mathematical Foundations of Quantum Mechanics 1252.2.9 Technical interemezzo: threeoperator topologies
In QM, there are at least 7 relevant topologies[ 13 ]which enter the game dis-
cussing sequences of operators. Here we limit ourselves to quickly illustrate the
three most important ones. We assume thatHis a complex Hilbert space though
the illustrated examples may be extended to more general context with some re-
adaptation.
(a)The strongest topology is theuniform operator topologyinB(H): It
is the topology induced by the operator norm|| ||defined in (2.29).
As a consequence of the definition of this topology, a sequence of elements
An∈B(H)issaidtouniformlyconverge toA∈B(H)when||An−A||→0for
n→+∞.
We already know thatB(H) is a Banach algebra with respect to that norm
and also aC∗algebra.
(b)IfL(D;H) withD⊂Ha subspace, denotes the complex vector space of the
operatorsA:D→H,thestrong operator topologyonL(D;H) is the topology
induced by the seminormspxwithx∈Dandpx(A):=||Ax||ifA∈L(D;H).
As a consequence of the definition of this topology, a sequence of elements
An∈L(D;H)issaidtostronglyconverge toA∈L(D;H)when||(An−A)x||→ 0
forn→+∞for everyx∈D.
It should be evident that, if we restrict ourselves to work inB(H), the uniform
operator topology is stronger than the strong operator topology.
(c)Theweak operator topologyonL(D;H) is the topology induced by the
seminormspx,ywithx∈H,y∈Dandpx,y(A):=|〈x, Ay〉|ifA∈L(D;H).
As a consequence of the definition of this topology, a sequence of elementsAn∈
L(D;H)issaidtoweaklyconverge toA∈L(D;H)when|〈x,(An−A)y〉|| → 0
forn→+∞for everyx∈Handy∈D.
It should be evident that, the strong operator topology is stronger than the
weak operator topology.
Example 2.2.72.
(1)Iff:R→Cis Borel measurable, andAa selfadjoint operator inH,consider
the sets
Rn:={r∈R||f(r)|<n} forn∈N.It is clear thatχRnf→fpointwise asn→+∞and that|χRnf|^2 ≤|f|^2 .Asa
consequence of restricting the operators on the left-hand side to Δf,
∫σ(A)χRnfdP(A)∣∣
∣∣
∣Δ
f→f(A) strongly, forn→+∞,