I'll answer the second part of the question.
I. Eigenvalues and Eigenfunctions
Let's first consider the one dimensional case n=1n=1. It is easy to check that the operator Rp1,p2Rp1,p2 has two eigenfunctions: 11 and
ξ(x)=(p1+p2)x−p1={−p1, if x=0,p2, if x=1.
ξ(x)=(p1+p2)x−p1={−p1,p2, if x=0, if x=1.
with eigenvalues
11 and
1−p1−p21−p1−p2, respectively.
Now consider the general case. For S⊂{1,…,n}S⊂{1,…,n}, let ξS(x)=∏i∈Sξ(xi)ξS(x)=∏i∈Sξ(xi). Observe that ξSξS is an eigenfunction of Rp1,p2Rp1,p2. Indeed since all variables xixi are independent, we have
Rp1,p2(ξ(x))=Rp1,p2(∏i∈Sξ(xi))=∏i∈SRp1,p2(ξ(xi))=∏i∈S((1−p1−p2)ξ(xi))=(1−p1−p2)|S|ξS(x).
Rp1,p2(ξ(x))=Rp1,p2(∏i∈Sξ(xi))=∏i∈SRp1,p2(ξ(xi))=∏i∈S((1−p1−p2)ξ(xi))=(1−p1−p2)|S|ξS(x).
We get that ξS(x)ξS(x) is an eigenfunction of Rp1,p2Rp1,p2 with eigenvalue (1−p1−p2)|S|(1−p1−p2)|S| for every S⊂{1,…,n}S⊂{1,…,n}. Since functions ξS(x)ξS(x) span the whole space, Rp1,p2Rp1,p2 has no other eigenfunctions (that are not linear combinations of ξS(x)ξS(x)).
II. Multiplicative Property
In general, the “multiplicative property” doesn't hold for Rp1,p2Rp1,p2 since the eigenbasis of Rp1,p2Rp1,p2 depends on p1p1 and p2p2. However, we have
R2p1,p2=Rp′1,p′2,
R2p1,p2=Rp′1,p′2,
where
p′1=2p1−(p1+p2)p1p′1=2p1−(p1+p2)p1 and
p′2=2p2−(p1+p2)p2p′2=2p2−(p1+p2)p2. To verify that, first note that
Rp1,p2Rp1,p2 and
Rp′1,p′2Rp′1,p′2 have the same set of eigenfunctions
{ξS}{ξS}. We have,
R2p1,p2(ξS)=(1−p1−p2)2|S|ξS=(1−p′1−p′2)|S|ξS=Rp′1,p′2(ξS)R2p1,p2(ξS)=(1−p1−p2)2|S|ξS=(1−p′1−p′2)|S|ξS=Rp′1,p′2(ξS)
since
1−p′1−p′2=1−p1⋅(2−(p1+p2))−p2⋅(2−(p1+p2))=1−(p1+p+2)(2−(p1+p2))=1−2(p1+p2)+(p1+p2)2=(1−p1−p2)2.1−p′1−p′2=1−p1⋅(2−(p1+p2))−p2⋅(2−(p1+p2))=1−(p1+p+2)(2−(p1+p2))=1−2(p1+p2)+(p1+p2)2=(1−p1−p2)2.
III. Relation to the Bonami—Beckner operator
Let us think of functions from {0,1}n{0,1}n to RR as polylinear polynomials. Let δ=12⋅p1−p2p1+p2δ=12⋅p1−p2p1+p2. Consider the operator
Aδ(f)=f(x1+δ,…,xn+δ).
Aδ(f)=f(x1+δ,…,xn+δ).
It maps every multilinear polynomial
ff to a multilinear polynomial
A[f]A[f]. We have,
Rp1,p2(f)=A−1δTεAδ(f),Rp1,p2(f)=A−1δTεAδ(f),
where
ε=1−p1−p2ε=1−p1−p2. Note that parts I and II follow from this formula and properties of the Bonami—Beckner operator.