I am a researcher whose research interest is in applying numerical algorithm to solve inverse problems. In reading the paper Global Uniqueness for a Two-Dimensional Inverse Boundary Value Problem, I have come across a notation issue. Specifically, let $\Omega$ be a subset of $\mathbb{R}^2$ with smooth boundary $\partial \Omega$. Let $ <\cdot,\cdot>$ be the bilinear pairing of the spaces $H^{1/2}(\partial \Omega)$ and $H^{-1/2}(\partial \Omega)$. Let $\Lambda_\gamma$ be the Dirichlet-to-Neumann (DtN) map associated to the partial differential equation
$$ \begin{cases} \nabla \cdot ( \gamma \nabla u) = 0 \\ u|_{\partial \Omega} = f \end{cases} $$
Explicitly, we have $ \Lambda_\gamma(f) = \left. \gamma \frac{\partial u}{\partial\nu} \right|_{\partial \Omega} $, where $\nu$ is the outward normal at the boundary $\partial \Omega$.
In Equation (0.21) of the paper, it reads $$ \left\langle g, \frac{\partial \gamma}{\partial \nu}f \right\rangle = \lim_{\substack{|\eta| \to \infty \\ \eta \in \mathbb{R} \times \{ 0 \}}} \langle g, e^{-i\langle \cdot , \eta \rangle} (\gamma \Lambda_1 + \Lambda_1 \gamma - 2 \Lambda_\gamma)e^{i\langle \cdot , \eta \rangle} f \rangle \tag{*} \label{eq:qn} $$ where $ f,g \in H^{1/2}(\partial \Omega)$ are continuous, $ i $ denotes the imaginary unit with $ i^2 = 1$ and $ \Lambda_1 $ is the DtN map with $ \gamma \equiv 1 $.
My main question is
What does the placeholder $ \cdot $ in \eqref{eq:qn} means?
The reason I am confused is that I don't see any operator on the left hand side, whereas I would expect placeholders are used in place of argument of a function or an operator.
