Permutation Operator

In the process of studying operator theory,researchers hope to have more concrete examples of operators using familiar mathematical objects.Let H be a separable Hilbert space with a fixed orthonormal basis {en}n∈A,where card(A)≤N0.Let S(A)be the permutation group of A,i.e.,the group of all bijections of A→A under function composition.For any σ∈S(A),define the bounded linear operator Tσ on H by Tσen=eσ(n),(?)n∈Z.This operator is called the permutation operator,and is unitary.The simplest example is the classical bilateral shift and identity operator.In Chapter 3,the product of disjoint permutations is defined,and it was proved that any(infinite)permutation a can be decomposed into a product of disjoint cyclesσ=Πi∈Γ(jki)k,where {jki}k is an indexed subset of Z and the cycle(jki)k represents the mapping that maps jki to jk+1i.For example,using σ=(...,-1,0,1,2,...)to represent the mapping that maps n to n+1 in Z,then σ3 maps n to n+3,which can be decomposed into three disjoint cycles,that is σ3=(...,-3,0,3,6,...)(...,-2,1,4,7,...)(...,-1,2,5,8,...).If we define τ(n)=-n,then we have τ=(0)(-1,1)(-2,2)...,and στ(n)=-n+1,στ=(1,0)(2,-1)(3,-2)....In Chapter 4,the method of type classification is used to show that the type function uniquely determines a permutation(up to similarity).Specifically,if σ∈S(A),defineμσ:{1,2,3,...}∪{∞}→{0,1,2,...}∪{∞},where μσ(n)is the number of n-cycles in the cycle decomposition of σ.Then,σ1∈S(A),σ2∈S(B)are similar if and only if σ1,σ2 have the same type.In this way,we uniquely determine a permutation by it’s type(up to similarity).In Chapter 5,the results above were used to study the properties of permutation operators.Specifically,for any Tσ,a reducing subspace decomposition is obtained as Tσ=⊕s∈Γ Ts,where T,is the bilateral shift or a cyclic matrix.A permutation operator is unitarily equivalent to its adjoint.If σ has no infinite orbits,then Tσ is diagonalizable.For the spectrum and numerical range,we have(1)σp(Tσ)=(?) is equivalent to μσ(n)=0,(?)n<∞.(2)Here,σap denotes the approximate point spectrum and T denotes the unit circle.(3)1 ∈σ(Tσ).(4)If σ does not have an infinite orbit,then the numerical range W(Tσ)is the convex hull of σp(Tσ).If a has an infinite orbit,then W(Tσ)=D∪σp(Tσ),where D denotes the open unit disk.For the topology and convergence of permutation operators,we have the following results:(1)Let σ,τ∈S(A)with σ≠τ.Then(?)≤‖Tσ-Tτ‖≤2.In this case,the norm topology on T(H)is the discrete topology,which means that T(H)is a discrete topological group with respect to the norm topology.(2)Let σ∈S(A)and {σi}i be a net of operators in S(A).The following statements are equivalent:1.Tσi(?)Tσ.2.Tσi(?)Tσ.3.For any n ∈ A,there exists i0 such that σi(n)=σ(n)for all i>i0.(3)The set of all permutation operators T(H)is not closed under the SOT or WOT topologies.T(H)is a topological group under the WOT topology.