Matrix coefficient of a unitary representation

Definition

Let $\pi:G\to U(\mathcal H)$ be a strongly continuous unitary representation on a complex Hilbert space, whose inner product is linear in the first variable. For $\xi,\eta\in\mathcal H$ , the matrix coefficient determined by $\xi$ and $\eta$ is

c_{\xi,\eta}(g)=\langle \pi(g)\xi,\eta\rangle,\qquad g\in G.

It is a bounded continuous complex-valued function, with $\lvert c_{\xi,\eta}(g)\rvert\leq \|\xi\|\,\|\eta\|$ . A coefficient with $\xi=\eta$ is called diagonal; it is positive-definite.

Basic properties

Coefficients are linear in $\xi$ and conjugate-linear in $\eta$ under the stated convention. Translation of the argument produces another coefficient: for $x\in G$ ,

c_{\xi,\eta}(xg)=c_{\xi,\pi(x^{-1})\eta}(g), \qquad c_{\xi,\eta}(gx)=c_{\pi(x)\xi,\eta}(g).

Thus the coefficient space attached to a representation is stable under left and right translation. The estimate in the core is the Cauchy–Schwarz inequality together with unitarity.

Structural role

Diagonal coefficients record the positive kernel associated with a vector, while polarization recovers every $c_{\xi,\eta}$ from diagonal coefficients. Conversely, the GNS construction for positive-definite functions realizes every continuous positive-definite function as a diagonal coefficient of a cyclic unitary representation. This makes coefficients the scalar observables used to compare representations without choosing operator coordinates Folland, §3.1–3.3.

Conventions and scope

If the inner product is taken linear in the second variable, the displayed formula remains meaningful but its linearity and the standard positive-definiteness convention must be adjusted consistently. The term representative function is often restricted to coefficients of finite-dimensional representations, so it should not be treated as an unrestricted synonym here.

References

G. B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: §§3.1–3.3 on unitary representations, coefficients, and functions of positive type.