the coupling vector $g$ belongs to the restricted parameter set $\Pi_r$ (\[Pir\]). This entails in particular that the integral operator ${{\cal I}}(g)$ (\[cI\]) is a well-defined HS operator on the Hilbert space ${{\cal H}}$ (\[cH\]). We begin this section by obtaining features of functions in the range of ${{\cal I}}(g)$.

Consider first for $f\in{{\cal H}}$ and $x\in[0,\pi/2r]$ the function $$\begin{gathered}
h(x)     \equiv    \int_0^{\pi/2r}dy{{\cal S}}(g;x,y)w(g';y)^{1/2}f(y)
\nonumber\\
\phantom{h(x)}{}   =       \int_0^{\pi/2r}dy\exp\left( 2s_g\sum_{n=1}^{\infty}\frac{\cos(2nrx)\cos(2nry)}{n\sinh(nr\alpha)}\right) w(g';y)^{1/2}f(y),\label{hdef}
  \end{gathered}$$ where we used (\[R2\]). Since we have $w(g';\cdot)^{1/2}\in{{\cal H}}$, we deduce from the Schwarz inequality $$\begin{gathered}
  w(g';y)^{1/2}f(y)\in L^1([0,\pi/2r],dy).
  \end{gathered}$$ It readily follows that $h(x)$ extends to a function $h(z)$, $\Re z=x$, that is analytic in the strip $|\Im z|<\alpha/2$. Also, $h(z)$ is $\pi/r$-periodic and even. Now $R(z)$ is an entire, $\pi/r$-periodic and even function without zeros for $|\Im z|<\alpha/2$, so the function $$\begin{gathered}
\label{ana}
  s(x)^{-g_0}s(\pi/2r-x)^{-g_1}({{\cal I}}(g) f)(x)=R(x)^{g_2}R(\pi/2r-x)^{g_3}h(x),\qquad x\in (0,\pi/2r),
  \end{gathered}$$ extends to a function that is analytic for $|\Im z|<\alpha/2$, $\pi/r$-periodic and even.

Next, we point out the relations $$\begin{gathered}
  g\in\Pi_r \ \Leftrightarrow \ g'\in\Pi_r, \nonumber
  \\
  \label{cIad}
  {{\cal I}}(g)^{*}={{\cal I}}(g'),
  \end{gathered}$$ which are easily verified. As a consequence, the orthocomplement of the range of ${{\cal I}}(g)$ equals the kernel of ${{\cal I}}(g')$. From (\[hdef\]) it is immediate that 