CLAIM-291: The adjoint of a square-integrable integral operator is the integral operator with the conjugate-transposed kernel

The claim

Let (X, μ) and (Y, ν) be σ-finite measure spaces and let k ∈ L²(μ⊗ν). The operator T : L²(ν) → L²(μ), (Tf)(x) = ∫ k(x,y) f(y) dν(y), is bounded with ‖T‖ ≤ ‖k‖₂. Its adjoint T* : L²(μ) → L²(ν) is

(T*g)(y) = ∫ k̄(x,y) g(x) dμ(x),

the integral operator with kernel k*(y,x) = k̄(x,y). For real kernels this is the transpose. When X = Y and μ = ν, T is self-adjoint if and only if k(x,y) = k̄(y,x) for μ⊗μ-almost every (x,y).

The argument

1. Integrability. For f ∈ L²(ν) and g ∈ L²(μ), Cauchy–Schwarz on X × Y gives ∫∫ |k(x,y)| |f(y)| |g(x)| dμ dν ≤ ‖k‖₂ ‖g ⊗ f‖₂ = ‖k‖₂ ‖f‖ ‖g‖ < ∞. The same bound with g ranging over the unit ball gives ‖T‖ ≤ ‖k‖₂.
2. Exchange. The integrand is therefore integrable, so Fubini applies: ⟨Tf, g⟩ = ∫ (∫ k(x,y) f(y) dν(y)) ḡ(x) dμ(x) = ∫ f(y) · conj(∫ k̄(x,y) g(x) dμ(x)) dν(y) = ⟨f, Sg⟩, with S the integral operator in the claim. S is bounded by step 1 applied to the kernel k̄(x,y) on Y × X.
3. Uniqueness. The adjoint is the unique operator with ⟨Tf, g⟩ = ⟨f, Tg⟩ for all f and g. Its existence for a bounded T is what the Riesz theorem supplies (LIT-230). Here it is exhibited directly. Two operators that satisfy the identity agree, because ⟨f, (S − S′)g⟩ = 0 for every f forces (S − S′)g = 0. So S = T.
4. Self-adjointness. With μ = ν, T = T* means the two kernels k and (x,y) ↦ k̄(y,x) define the same operator. The map from L²(μ⊗μ) kernels to Hilbert–Schmidt operators is an isometry, so it is injective, and the kernels agree almost everywhere.

This is the textbook fact for Hilbert–Schmidt operators. The record holds no reading of a textbook that proves it, so the proof above is the record’s own.

Where it is used

• THEORY-009. It says HaoChen et al. apply the spectral theorem to a square-integrable kernel k(u,v) = w(u,v)/√(w(u)w(v)). By step 4 that operator is self-adjoint because w(u,v) = w(v,u). This claim is the step THEORY-009 marks as not the Riesz article’s.
• The X ≠ A case. With two spaces (Zhai’s thesis, LIT-975, Theorem 2.11) the formula still gives the adjoint. TT and TT are then self-adjoint whatever k is. That is the composition THEORY-009 records as the thesis’s source of self-adjointness.

What it does not say

• Nothing about kernels outside L²(μ⊗ν). For a bounded but not Hilbert–Schmidt kernel operator the same formula holds when the integrand is integrable, for instance under a Schur test. That case is not argued here.
• Nothing about whether a self-supervised pair distribution makes k symmetric. That is THEORY-009’s claim, and it rests on the readings it cites, not on this one.