Convolution operators

Assume the Axiom of Choice. Let $G$ be a compact Lie group with normalized Haar measure $dg$, and let $L^2(G)$ be the complex Hilbert space of the left and right regular representations (Left and right regular representations on L2(G)). For a continuous function $k\in C(G,\mathbb{C})$ the convolution operator with kernel $k$ is $(T_{k}f)(x):={\int }_{G}k(x^{-1}y)f(y)dy,x\in G,f\in L^2(G)\mathrm{.}$ This is the right-convolution convention fixed for the whole page. The integral converges absolutely for every $f\in L^2(G)$ by Cauchy–Schwarz, because $k$ is bounded and $G$ has finite measure, and $T_{k}f$ is a well-defined element of $L^2(G)$: the bound $\mathrm{\parallel }{T}_{k}f{\mathrm{\parallel }}_2\le \mathrm{\parallel }k{\mathrm{\parallel }}_2\mathrm{\parallel }f{\mathrm{\parallel }}_2$ is proved together with the Hilbert–Schmidt property on this page. The assignment $f\mapsto T_{k}f$ is linear, so $T_k$ is a bounded linear operator on $L^2(G)$ with $\mathrm{\parallel }{T}_k\mathrm{\parallel }\le \mathrm{\parallel }k{\mathrm{\parallel }}_{\mathrm{\infty }}$ and also $\mathrm{\parallel }{T}_k\mathrm{\parallel }\le \mathrm{\parallel }k{\mathrm{\parallel }}_2$.

The right translate and left translate of a kernel are $k_h^{\mathrm{R}}(x):=k(xh),k_h^{\mathrm{L}}(x):=k(h^{-1}x),$ and the adjoint kernel is $k^{\ast }(x):=\overline{k(x^{-1})}$.

• The convention $(T_{k}f)(x)={\int }_{G}k(x^{-1}y)f(y)dy$ makes $T_k$ the operator associated with right translation of the argument: after the substitution $y=xu$ and use of left invariance of Haar measure, $T_{k}f(x)={\int }_{G}k(u)f(xu)du\mathrm{.}$ In general this is not ${\int }_{G}f(y)k(y^{-1}x)dy$; that expression uses the inverted kernel and agrees with this convention only under an additional inversion symmetry of $k$.
• The kernel $K(x,y)=k(x^{-1}y)$ of $T_k$ is continuous on $G\times G$; it is the kernel whose square-integrability is proved on this page.
• Convolution is commutative on central functions, and for $k$ central the operator $T_k$ commutes with both regular actions; this is used in the approximate-identity and Peter–Weyl arguments.