PRELIMINARIES AND NOTATION bounded,ic, le(u)l s Cllull for all n E H, there exists a unique ve E H so that e(u=(u, ve An operator U from H, to H2 is an isometry if (Uv, Uw)=u, w) for all U,t∈H1,U itary if moreover UHI= 02, i.e every element u2 E H2 can be written as v?= Uu, for some v1 E H,. If the en constitute an orthonormal basis in HI, and U is unitary, then the Uen constitute an orthonormal basis in H2. The reverse is also true any operator that maps an orthonormal basis to another orthonormal basis is unitary a set D is called dense in H if every u E 7 can be written as the limit of some sequence of un in D.(One then says that the closure of D is all of H. The closure of a set s is obtained by adding to it all the v that can be obtained as limits of sequences in S. )If Au is only defined for v E D, but we know that A≤Cl‖ for all 1∈D (0.09) then we can extend a to all of H "by continuity Explicitly: if u E H, find un E D so that limn-go n=1. Then the un are necessarily a Cauchy sequence, and because of (0.0.9 ), so are the Aun; the Aun have therefore a limit, which we call Au(it does not depend on the particular sequence un that was chosen) One can also deal with Inbounded operators, i.e., A for which there exists no finite C such that‖Al≤Clu‖ holds for all 1∈. It is a fact of life that these can usually only be defined on a dense set D in H, and cannot be extended with compact support, for D. The dense set on which the operator is defined is called its domain I ne adjoint A'of a bounded operator A from a Hilbert, space H, to a Hilbert space H2(which may be H1 itself)is the operator from H2 to Hi defined by (u1, Au2)=(A which should hold for all uy∈化1,u2∈2.( The existence of A'is guaranteed by Riesz'representation theorem: for fixed u2, we can define a linear functional e on Hl by e(u1)=(Au1, u2). It is clearly bounded, and corresponds therefore to a vector v so that(ul, v)=e(u). It is easy to check that the correspondence u2+v is linear; this defines the operator A. )One has A’‖=‖A|,‖AAI=‖A|2 If A'=A(only possible if A maps H to itself), then A is called self-adjoint. If a self-adjoint operator A satisfies( Au, u)>0 for all u E H, then it is' called a positive operator; this is often denoted A>0. We will write A> B if'A-B is Trace-class operators are special operators such that 2n I(Aen, en)I is finite for all orthonormal bases in H. For such a trace-class operator, >n(Aen, en)is independent of the chosen orthonormal basis; we call this sum the trace of A A=∑(Ae