Hardy space infinite elements for time harmonic wave equations with phase and group velocities of different signs

Abstract

We consider time harmonic wave equations in cylindrical wave-guides with physical solutions for which the signs of group and phase velocities differ. The perfectly matched layer methods select modes with positive phase velocity, and hence they yield stable, but unphysical solutions for such problems. We derive an infinite element method for a physically correct discretization of such wave-guide problems which is based on a Laplace transform in propagation direction. In the Laplace domain the space of transformed solutions can be separated into a sum of a space of incoming and a space of outgoing functions where both function spaces are Hardy spaces of a curved domain. The Hardy space is constructed such that it contains a simple and convenient Riesz basis with small condition numbers. In this paper the new method is only discussed for a one-dimensional fourth order model problem. Exponential convergence is shown. The method does not use a modal separation and works on an interval of frequencies. Numerical experiments confirm exponential convergence.

Appendix

1.1 Hardy spaces

In this section we collect the definitions and the required properties of standard and non-standard Hardy spaces. Proofs which are not carried out here can be found in [6] or [11]. We denote complex contour integrals by $\int ds$ and line integrals by $\int |ds|$.

Definition A.1

(Hardy space on the unit disk and its complement) Let $S^1:=\{z \in \mathbb {C}:|z|=1\}$ denote the unit circle. The Hardy spaces $H^{\pm }(S^1)$ are the sets of all functions $U_{bd}\in L^2(S^1)$ for which there exists a holomorphic function $U_{vol}$ on $\{z \in \mathbb {C}:|z|^{\pm 1}<1\}$ such that

$$\begin{aligned}&\sup _{0<r<1}\int _0^{2\pi }|U_{vol}(r^{\pm 1}e^{i\theta })|^2d\theta <\infty \quad \text{ and }\quad \lim _{r\nearrow 1}\int _0^{2\pi }|U_{bd}(i\theta )U_{vol}(r^{\pm 1}e^{i\theta })|^2d\theta = 0.\nonumber \\ \end{aligned}$$

(A.1)

We say that $U_{bd}$ are $L^2$-boundary values of $U_{vol}$.

It can be shown that $U_{vol}$ is uniquely determined by $U_{bd}$ and can be computed by Cauchy’s integral formula. This one-to-one correspondence of $U_{vol}$ and $U_{bd}$ explains the terminology. Note that for the Hardy space $H^+(S^1)$ on the unit disk $\{z \in \mathbb {C}:|z|<1\}$ the first condition in (A.1) is not required.

Lemma A.2

(Properties of $H^+(S^1))$ Equipped with the $L^2$-inner product, $H^+(S^1)$ is a Hilbert space. A complete orthogonal basis of $H^+(S^1)$ is given by the monomials $z^n, n = 0, 1, \ldots$. Moreover, $L^2(S^1)=H^+(S^1)\oplus ^{\perp }H^-(S^1)$.

For $\kappa _0\in \mathbb {C}{\setminus }\{0\}$ the Moebius transformation $m_{\kappa _0}:\mathbb {C}{\setminus }\{1\} \rightarrow \mathbb {C}$ is defined by $m_{\kappa _0}(z):=i\kappa _0 \frac{z+1}{z-1}$. Note that $m_{\kappa _0}(S^1{\setminus }\{1\})= \kappa _0\mathbb {R}$ with $\kappa _0\mathbb {R}:=\{\kappa _0 x :x \in \mathbb {R}\}$. Therefore, we can define the pull-back operators

$$\begin{aligned} {\mathscr {M}}_{\kappa _0}:L^2(\kappa _0\mathbb {R})\rightarrow L^2(S^1), \qquad ({\mathscr {M}}_{\kappa _0} f)(z):=\tfrac{(f \circ m_{\kappa _0})(z)}{z-1}, \end{aligned}$$

where the weight factor $\frac{1}{z-1}$ is chosen such that ${\mathscr {M}}_{\kappa _0}$ is well-defined and unitary up to the factor $\sqrt{-2i\kappa _0}$.

Definition A.3

(Hardy space on half planes) For $\kappa \in \mathbb {C}{\setminus }\{0\}$ the Hardy spaces $H^\pm (\kappa _0\mathbb {R})$ on the half-planes $\{\kappa _0(x\pm iy):x\in \mathbb {R}, y>0\}$ are defined by $H^\pm (\kappa _0\mathbb {R}):={\mathscr {M}}_{\mp \kappa _0}^{-1} H^+(S^1)$. ($\kappa _0\mathbb {R}$ has to be considered as an oriented curve to distinguish between these spaces!) They are equipped with the inner product $(f,g)_{H^\pm (\kappa _0\mathbb {R})}:=\int _{\kappa _0\mathbb {R}} f(s) \overline{g(s)} |ds|$. For $\kappa _0=1$ we will omit the parameter and shortly write $H^{\pm }(\mathbb {R})$.

The definition differs from the one given in most text books where $H^-(\mathbb {R})$ is characterized as the set of all functions $U_{bd}\in L^2(\mathbb {R})$ that are $L^2$ boundary values of a function $U_{vol}$ which is analytic in $\mathbb {C}^-:=\{z\in \mathbb {C}:\mathfrak {I}z<0\}$ and for which the integrals $\int _{\mathbb {R}}|U_{vol}(x-iy)|^2dx$ are uniformly bounded for $y \in (0,\infty )$. Due to [11] it is clear that these definitions are equivalent. We stick to Definition A.3 because it is easier to generalize to other boundaries.

Lemma A.4

(Properties of $H^-(\mathbb {R}))$

• (1) $U_{bd}\in L^2(\mathbb {R})$ belongs to $H^-(\mathbb {R})$ if and only if there exists an analytic function $U_{vol}$ in $\mathbb {C}^-$ and a sequence of rectifiable Jordan curves $C_1,C_2,\dots$ tending to the boundary in $\mathbb {C}^-$ such that the integrals $\int _{C_n} |U_{vol}(z)|^2|dz|$ are uniformly bounded and $U_{vol}(s)\rightarrow U_{bd}(t)$ for $s\in \mathbb {C}^-,s\rightarrow t$ non-tangentially for almost all $t\in \mathbb {R}$. With $C_n$ tending to the boundary we mean that $C_n$ eventually surrounds each compact subset of $\mathbb {C}^-$.

• (2) $\mathscr {M}_1$ is unitary from $H^-(\mathbb {R})$ to $H^+(S^1)$.

• (3) Equipped with the $L^2(\mathbb {R})$-inner product $H^-(\mathbb {R})$ is a Hilbert space with orthonormal basis $\frac{2i}{s-i}\left( \frac{s+i}{s-i}\right) ^n$, $n=0,1,\dots$.

• (4) $L^2(\mathbb {R})=H^+(\mathbb {R})\oplus ^{\perp }H^-(\mathbb {R})$.

• (5) Let $\varLambda \subset \mathbb {C}^+:=\{s \in \mathbb {C}:\mathfrak {I}s > 0\}$ be an infinite set with a cluster point in $\mathbb {C}^+$. Then $H^-(\mathbb {R}) = \overline{ {\text {span}}\{ \frac{1}{\bullet -\lambda }:\lambda \in \varLambda \} }^{H^-(\mathbb {R})}$.

Proof

The first four points follow from [6, Sect. 11, Example 1] and the definition of $H^-(\mathbb {R})$. The last one can be found in [12, Lemma A.2]. $\square$

Assumption A.5

Let $\varGamma =\gamma (\mathbb {R})$ be an oriented curve parameterized by a twice continuously differentiable function $\gamma :\mathbb {R}\rightarrow \mathbb {C}$ of the form

$$\begin{aligned} \gamma (\rho ) = \rho \sigma (|\rho |),\qquad \rho \in \mathbb {R}\end{aligned}$$

(A.2)

with $\sigma :[0,\infty )\rightarrow S^1$, and suppose that $\sup _{\rho \in \mathbb {R}}|\gamma '(\rho )|<\infty$ and there exists $\sigma _{\infty }\in S^1$ such that $\lim _{|\rho |\rightarrow \infty }|\gamma (\rho )-\sigma _{\infty }\rho |=0$.

Note that any curve $\varGamma$ satisfying Assumption A.5 is point symmetric, i.e. $-\varGamma =\varGamma$ and separates $\mathbb {C}$ into the unbounded, simply connected sets

$$\begin{aligned} \varGamma ^{\pm }:=\{\gamma (\rho )\exp (\pm it): \rho >0, t\in (0,\pi )\}, \end{aligned}$$

(A.3)

i.e. $\mathbb {C}=\varGamma ^-\dot{\cup }\varGamma \dot{\cup }\varGamma ^+$.

Lemma A.6

If $\varGamma$ satisfies Assumption A.5, there exists a bijective, continuously differentiable mapping $\eta :\mathbb {C}^-\cup \mathbb {R}\rightarrow \varGamma ^-\cup \varGamma$ such that $\eta$ is conformal on $\mathbb {C}^-$, $\eta (\mathbb {R})=\varGamma$ and $\log$ can be defined analytically on $\eta '(\mathbb {C}^-)$.

Proof

Due to the Riemann mapping theorem, see e.g. [3] there exists a conformal bijective mapping $\eta :\mathbb {C}^-\rightarrow \varGamma ^-$. Since $\mathbb {C}^-$ is simply connected, so is $\eta '(\mathbb {C}^-)$. As $\eta$ is conformal, we have $0\notin \eta '(\mathbb {C}^-)$, and the logarithm can be defined analytically. We need to show that $\eta$ can be extended with the stated properties. If $\varGamma ^-$ would be bounded, [16, Sect. 3.3] would already give us the claimed extension. As the stated properties are of local nature, we can generalize the results in [16, Sect. 3.3]. $\square$

Definition A.7

Let $\varGamma$ fulfill Assumption A.5 and let $\eta$ be a mapping as described in Lemma A.6. Let ${\mathscr {N}}_{\eta }: L^2(\varGamma ) \rightarrow L^2(\mathbb {R})$ with $({\mathscr {N}}_{\eta }f)(s):=(f\circ \eta )(s)\sqrt{\eta '(s)}$ for $s\in \mathbb {R}$ and $f \in L^2(\varGamma )$, and ${\mathscr {N}}_{\eta }^{-1}:L^2(\mathbb {R}) \rightarrow L^2(\varGamma )$ with ${\mathscr {N}}_{\eta }^{-1}g=\frac{g}{\sqrt{\eta '}}\circ \eta ^{-1}$ for $g\in L^2(\mathbb {R})$.

Then we define the Hardy space $H(\varGamma )^{\pm }$ by $H(\varGamma )^{\pm }:={\mathscr {N}}_{\mp \eta }^{-1}H^-(\mathbb {R})$ with the inner product $(f,g)_{H^-(\varGamma )}=\int _{\varGamma }f(s)\bar{g}(s)|ds|$.

Due to Lemma A.6 ${\mathscr {N}}_{\eta }$ and ${\mathscr {N}}_{\eta }^{-1}$ are well defined. $H^{-}(\varGamma )$ is independent of the choice of $\eta$ since for two such mappings $\eta _1$ and $\eta _2$ the composition $\eta _2^{-1}\circ \eta _1:\mathbb {C}^-\rightarrow \mathbb {C}^l$ is an automorphism of $\mathbb {C}^-$ with $\infty$ as invariant point, and $H^-(\mathbb {R})$ is invariant under such transformations.

Lemma A.8

(Properties of $H^-(\varGamma ))$ Let Assumption A.5 hold true. Then

• (1) $U_{bd}\in L^2(\varGamma )$ belongs to $H^-(\varGamma )$ if and only if there exists an analytic function $U_{vol}$ in $\varGamma ^-$ and a sequence of rectifiable Jordan curves $C_1,C_2,\dots$ tending to the boundary in $\varGamma ^-$ such that the integrals $\int _{C_n} |U_{vol}(s)|^2|ds|$ are uniformly bounded and $U_{vol}(s)\rightarrow U_{bd}(t)$ for $s\in \varGamma ^-,s\rightarrow t$ non-tangentially for almost all $t\in \varGamma$.

• (2) ${\mathscr {N}}_{\eta }$ is an isometry between $H^-(\varGamma )$ and $H^-(\mathbb {R})$.

• (3) Equipped with the $L^2(\varGamma )$ inner product $H^-(\varGamma )$ is a Hilbert space.

• (4) $\frac{1}{\bullet -\lambda }\in H^-(\varGamma )$ for all $\lambda \in \varGamma ^+$.

• (5) For $U_{bd}\in H^-(\varGamma )$ there holds $U_{vol}(z)=\frac{1}{2 \pi i}\int _{\varGamma }\frac{U_{bd}(s)}{s-z}ds$ for all $z\in \varGamma ^-$ and $\frac{1}{2 \pi i}\int _{\varGamma }\frac{U_{bd}(s)}{s-z}ds=0$ for all $z\in \varGamma ^+$.

Proof

The first three points are obvious. (4) We use A.8(1) as characterization of $H^-(\varGamma )$. Since $\frac{1}{s-\lambda }$ is analytic in $\varGamma ^-$, we need to state a sequence of rectifiable Jordan curves $C_n$ tending to the boundary in $\varGamma ^-$ such that the integrals $\int _{C_n} \frac{1}{|z-\lambda |^2}|dz|$ are uniformly bounded. Such curves can be constructed by choosing

$$\begin{aligned} \gamma _n(\rho ):=\gamma (\rho )\exp \left( -\frac{i}{n}\frac{\rho }{1+\rho ^2}\right) ,\qquad \rho \in \mathbb {R},\quad n\in \mathbb {N}. \end{aligned}$$

The curves $C_n^{(1)}:=\gamma _n([-n,n])$ are closed by circular arcs $C_n^{(2)}:=\{\gamma (n)\exp (-it):\frac{n}{1+n^2}\le t\le \pi -\frac{n}{1+n^2}\}$, i.e. $C_n:=C_n^{(1)}\cup C_n^{(2)}$. It is easy to see that $C_n\subset \varGamma ^-$ and that the curves $C_n$ eventually surround any compact subset of $\varGamma ^-$. The integrals $\int _{C_n^{(1)}}|s-\lambda |^{-2}\,|ds| = \int _{-n}^n|\gamma _n(\rho )-\lambda |^{-2}|\gamma _n'(\rho )|d\rho$ are uniformly bounded since $\sup _{n\in \mathbb {N}}|\gamma _n(\rho )-\lambda |^{-2}=O(\rho ^{-2})$ for $|\rho |\rightarrow \infty$, $\sup _{n\in \mathbb {N},\rho \in \mathbb {R}} |\gamma _n(\rho )-\lambda |^{-2}<\infty$, and $\sup _{n\in \mathbb {N},\rho \in \mathbb {R}}|\gamma _n'(\rho )| \le \sup _{n\ge n_0,\rho \in \mathbb {R}}|\gamma '(\rho )|+|\gamma (\rho )|^2 n^{-1}(1-\rho ^2)(1+\rho ^2)^{-2}<\infty .$ Similarly, it is easy to see that $\int _{C_n^{(2)}}|s-\lambda |^{-2}\,|ds|$ is uniformly bounded.

(5) Let $z\in \varGamma ^-$ be given. Using the mapping $\eta :\mathbb {R}\rightarrow \varGamma$ defined in Lemma A.6 Cauchy’s integral Theorem leads together with Lemma A.8 to

$$\begin{aligned} U_{vol}(z)&=\lim _{y \nearrow 0} \lim _{R \rightarrow \infty } \frac{1}{2\pi i} \int _{\eta \left( [-R+iy,R+iy]\cup \{R\exp (i[0,\pi ])+iy\}\right) }\frac{U_{vol}(s)}{s-z}ds\\&=\frac{1}{2\pi i} \lim _{y \nearrow 0} \int _{\mathbb {R}} ({\mathscr {N}}_{\eta } U_{vol})(x+iy) \left( {\mathscr {N}}_{\eta } \left\{ (\bullet -z)^{-1}\right\} \right) (x+iy)\,dx\\&= \frac{1}{2\pi i} \int _{\mathbb {R}} ({\mathscr {N}}_{\eta } U_{bd})(x) \left( {\mathscr {N}}_{\eta } \left\{ (\bullet -z)^{-1}\right\} \right) (x)\,dx = \frac{1}{2 \pi i}\int _{\varGamma }\frac{U_{bd}(s)}{s-z}ds. \end{aligned}$$

The second statement follows analogously since for $z\in \varGamma ^+$ the integrand is holomorphic in $\varGamma ^-$. $\square$

Lemma A.9

Let $\varGamma$ satisfy Assumption A.5.

• (1) The generalized Hilbert transform ${{\mathscr {H}}}_{\varGamma }:L^2(\varGamma )\rightarrow L^2(\varGamma )$,

  $$\begin{aligned} \left( {{\mathscr {H}}}_{\varGamma }U\right) (z) := P.V.\frac{1}{\pi i}\int _{\varGamma }\frac{U(s)}{s-z}ds :=\lim _{\epsilon \searrow 0, R\rightarrow \infty }\left( \frac{1}{\pi i} \int _{(\varGamma \cap B_{R}(z){\setminus } B_{\epsilon }(z)}\frac{U(s)}{s-z}ds\right) \end{aligned}$$

  is a well-defined bounded linear operator.

• (2) For every $U\in L^2(\varGamma )$ the Sokhotski-Plemelj jump relations

  $$\begin{aligned} \displaystyle \lim _{\lambda \rightarrow z,\lambda \in \varGamma ^{\pm }}\frac{1}{2\pi i}\int _{\varGamma }\frac{U(s)}{s-\lambda }ds=\frac{1}{2}\left( U(z)\mp {{\mathscr {H}}}_{\varGamma }U(z)\right) , \quad z\in \varGamma \end{aligned}$$

  hold true in the $L^2$ sense (see Lemma A.8.)

• (3) The operators ${\mathscr {P}}^{\pm }:=\frac{1}{2}(\mathrm id\mp {{\mathscr {H}}}_{\varGamma })$ are bounded linear projections of $L^2(\varGamma )$ onto $H^{\pm }(\varGamma )$ along $H^{\mp }(\varGamma )$. In particular, we have the topological sum

  $$\begin{aligned} L^2(\varGamma )=H^{+}(\varGamma )\oplus H^{-}(\varGamma ). \end{aligned}$$

Proof

(1) Since $1\le |\gamma '(\rho )|\le C<\infty$ for all $\rho \in \mathbb {R}$, the pull-back $L^2(\varGamma )\rightarrow L^2(\mathbb {R})$, $U\mapsto \widetilde{U}:= U\circ \gamma$ is bounded and boundedly invertible. Therefore, it suffices to show that the linear operator $\tilde{{{\mathscr {H}}}_{\varGamma }}:L^2(\mathbb {R})\rightarrow L^2(\mathbb {R})$, $\tilde{U}\mapsto ({{\mathscr {H}}}_{\varGamma }U)\circ \gamma$ is well-defined and bounded. In the numerator of the kernel of ${{\mathscr {H}}}_{\varGamma }$ we add and subtract a smooth, symmetric cut-off function $\chi :\mathbb {R}\rightarrow [0,1]$ satisfying $\chi (\rho )=1$ for $|\rho |\le 1$ and $\chi (\rho )=0$ for $|t|\ge 2$. Then

$$\begin{aligned} \pi i(\widetilde{{{\mathscr {H}}}_{\varGamma }}\widetilde{U})(r)= & {} P.V.\int _{\mathbb {R}}\frac{\chi (\rho -r)}{\gamma (\rho )-\gamma (r)} \widetilde{U}(\rho ) \gamma '(\rho )d\rho \\&+ P.V.\int _{\mathbb {R}}\frac{1-\chi (\rho -r)}{\gamma (\rho )-\gamma (r)} \widetilde{U}(\rho ) \gamma '(\rho )d\rho \\=: & {} (T_1\widetilde{U})(r)+(T_2\widetilde{U})(r). \end{aligned}$$

To show $L^2$-boundedness of $T_1$ we will use smoothness of $\varGamma$ and for $L^2$-boundedness of $T_2$ the behavior of $\varGamma$ at infinity. We split off the singularity of $T_1$ as a convolution operator by writing $\gamma (\rho )-\gamma (r) = \gamma '(\rho )(\rho -r+R(\rho ,r))$ with the scaled Taylor remainder $R(\rho ,r):=\frac{\gamma (\rho )-\gamma (r)}{\gamma '(\rho )}-(\rho -r)$. Then

$$\begin{aligned} (T_1\widetilde{U})(r)= & {} P.V.\int _{|\rho -r|\le 2}\frac{\chi (\rho -r)}{\rho -r} \widetilde{U}(\rho )d\rho \nonumber \\&- \int _{|\rho -r|\le 2} \frac{\chi (\rho -r)R(\rho ,r)}{(\rho -r)(\gamma (\rho )-\gamma (r))} \gamma '(\rho )\widetilde{U}(\rho )d\rho . \end{aligned}$$

(A.4)

The $L^2$-boundedness of the convolution operator can e.g. be derived from the $L^2$-boundedness of the classical Hilbert transform since the Fourier transforms of the convolution kernel is a convolution of the (distributional) Fourier transform of Hilbert transform kernel $t\mapsto 1/t$ and the Fourier transform of $\chi$. Hence, we have the Fourier transform of an $L^{\infty }$ and an $L^1$-function, which belongs to $L^{\infty }$. As $|\gamma (\rho )-\gamma (r)|\ge |\rho -r|$ and $|R(\rho ,r)|\le C|\rho -r|^2$ with a constant $C>0$ independent of t, r with $|\rho -r|\le 2$, the kernel of the second integral operator is uniformly bounded. Together with boundedness of the integration domain this easily yields $L^2$-boundedness.

The operator $T_2$ can be decomposed into

$$\begin{aligned} (T_2\widetilde{U})(r)= & {} P.V.\int \limits _{|\rho -r|\ge 1}\frac{1-\chi (\rho -r)}{\sigma _{\infty } \rho -\sigma _{\infty } r}\gamma '(\rho )\widetilde{U}(\rho )d\rho \\&-\int \limits _{|\rho -r|\ge 1}(1-\chi (\rho -r)) \frac{(\gamma (\rho )-\sigma _{\infty } \rho )-(\gamma (r)-\sigma _{\infty } r)}{(\gamma (\rho )-\gamma (r))(\rho -r)\sigma _{\infty }} \gamma '(\rho )\widetilde{U}(\rho )d\rho . \end{aligned}$$

As the first integral equals $\frac{1}{\sigma _{\infty }}P.V.\int \frac{1}{\rho -r}\gamma '(\rho )\widetilde{U}(\rho )d\rho -\frac{1}{\sigma _{\infty }}P.V.\int \frac{\chi (\rho -r)}{\rho -r}\gamma '(\rho )\widetilde{U}(\rho )d\rho$, it defines a linear operator on $L^2(\mathbb {R})$ due to the $L^2$ boundedness of the convolution operator in (A.4) and boundedness of the classical Hilbert transform. As $\sup _{\rho \in \mathbb {R}}|\gamma (\rho )-\sigma _{\infty } \rho |<\infty$ by Assumption A.5 and $|\gamma (\rho )-\gamma (r)|>|\rho -r|$, the kernel $k(\rho ,r)$ of the second integral operator in the decomposition of $T_2$ satisfies $|k(\rho ,r)|\le C|\rho -r|^{-2}$ for some $C>0$ and all $|\rho -r|\ge 1$. This bound implies $L^2$-boundedness since by the Cauchy-Schwarz inequality and the identity $\int _{|\rho |\ge 1}\frac{1}{\rho ^2}d\rho =2$ we have

$$\begin{aligned} \int \left| \int _{|\rho -r|\ge 1} k(\rho ,r)\widetilde{U}(\rho )d\rho \right| ^2 dr&\le C^2\int \int _{|\rho -r|\ge 1}\frac{1}{(\rho -r)^2} d\rho \int _{|\rho -r|\ge 1} \frac{|\widetilde{U}(\rho )|^2}{(\rho -r)^2} d\rho \,dr\\&= 2C^2 \int \int _{|\rho -r|\ge 1}\frac{1}{(\rho -r)^2}dr |\widetilde{U}(\rho )|^2d\rho \\&=4C^2\Vert \widetilde{U}\Vert _{L^2}^2. \end{aligned}$$

(2) This easily follows from the corresponding result for closed curves in [3] using a partition of unity.

(3) It follows from part 2 that ${\mathscr {P}}^{\pm } U\in H^{\pm }(\varGamma )$ for all $U\in L^2(\varGamma )$. Together with Lemma A.8 we obtain ${\mathscr {P}}^{\pm }U_\pm =U_\pm$ and ${\mathscr {P}}^{\mp }U_\pm =0$ for $U_\pm \in H^{\pm }(\varGamma )$. $\square$

Lemma A.10

Let Assumption A.5 hold true and let $\varLambda ^{\pm } \subset \varGamma ^\pm$ be two infinite sets with cluster points in $\varGamma ^\pm$. Then $H^\mp (\varGamma ) = \overline{{\text {span}} \{\frac{1}{\bullet -\lambda }:\lambda \in \varLambda ^{\pm } \}}^{H^\mp (\varGamma )}$.

Proof

Let $\varLambda ^{\pm } \subset \varGamma ^{\pm }$ be two infinite sets with cluster points in $\varGamma ^{\pm }$. We prove that ${\text {span}} \{(\bullet -\lambda )^{-1}: \lambda \in \varLambda ^+\cup \varLambda ^-\}$ is dense in $L^2(\varGamma )$. Then the assertion follows with A.9(3) and $(\bullet -\lambda )^{-1} \in H^\mp (\varGamma )$ for $\lambda \in \varLambda ^\pm$ due to A.8(4).

Take $U\in L^2(\varGamma )$ such that $U\perp _{L^2(\varGamma )}(s-\lambda )^{-1}$ for all $\lambda \in \varLambda ^+\cup \varLambda ^-$. Hence with the tangent function $\mathfrak {t}(s)\ne 0$ there holds $\int _{\varGamma } \overline{U(s)\mathfrak {t}(s)}(s-\lambda )^{-1}ds=0$ for all $\lambda \in \varLambda ^+\cup \varLambda ^-$. Hence, the function $g:\varLambda ^+\cup \varLambda ^- \rightarrow \mathbb {C}$ with $g(\lambda ):=\int _{\varGamma } \overline{U(s)\mathfrak {t}(s)}(s-\lambda )^{-1}ds$ is holomorphic in $\varGamma ^{+}\cup \varGamma ^{+}$ and vanishes in $\varLambda ^+\cup \varLambda ^-$. Therefore it vanishes in $\mathbb {C}{{\setminus }} \varGamma$ and with A.9(2) it holds $U=0$.$\square$

1.2 Toeplitz operators

When working with operator theory on Hardy spaces, Toeplitz operators are a most natural class of operators to deal with. From the rich theory of Toeplitz operator we only need a few rather simple results, which can all be found, e.g., in [5].

Definition A.11

(Multiplication and Toeplitz operator) Let H be a Hilbert space and $\mathscr {B}(H)$ the set of bounded linear operators on H. For a function $a\in L^{\infty }(S^1)$ we define the multiplication operator ${\mathscr {Q}}(a)\in \mathscr {B}(L^2(S^1))$ point-wise by $\left( {\mathscr {Q}}(a) u\right) (z):= a(z) u(z)$ for all $z \in S^1$ and $u \in L^2(S^1)$.

Moreover, if $\mathscr {P}$ denotes the orthogonal projection from $L^2(S^1)$ to $H^+(S^1)$, the Toeplitz operator ${\mathscr {T}}(a)\in \mathscr {B}(H^+(S^1))$ with symbol a is defined by ${\mathscr {T}}(a):=\mathscr {P}{\mathscr {Q}}(a)$.

Definition A.12

(Toeplitz matrix) An infinite matrix $T\in \mathscr {B}(l^2(\mathbb {N}_0))$, $(Tf)_n:= \sum _{m=0}^\infty T_{n,m}f_m$ is called an (infinite) Toeplitz matrix if $T_{n,m}=T_{n+1,m+1}$ for all $n,m\in \mathbb {N}_0$. For a Toeplitz operator ${\mathscr {T}}(a):H^+(S^1) \rightarrow H^+(S^1)$ with symbol $a\in L^{\infty }(S^1)$ the associated infinite Toeplitz matrix T(a) is given as $T(a)_{n,m}:=\int _{S^1}a(z)z^{n-m}|dz|$ for all $n,m \in \mathbb {N}_0$.

Note that T(a) is the matrix representation of ${\mathscr {T}}(a)$ with respect to the orthonormal basis $\{z^k,~k\in \mathbb {N}_0\}$ of $H^+(S^1)$.

Vice versa, let $T\in \mathscr {B}(l^2(\mathbb {N}_0))$ be an infinite Toeplitz matrix. Note that $T_{n,m}$ only depends on the difference of the indices, i.e. $T_{n,m}=a_{n-m}$, $n,m\in \mathbb {N}_0$ for some numbers $a_k\in \mathbb {C}$, $k\in \mathbb {Z}$. Then it is easy to show that $T=T(a)$ is the Toeplitz matrix associated to the Toeplitz operator ${\mathscr {T}}(a):H^+(S^1) \rightarrow H^+(S^1)$ with symbol $a\in L^{\infty }(S^1)$ given by $a(z):=\sum _{k\in \mathbb {Z}}a_kz^k$, $z \in S^1$.

Due to this one-to-one correspondence of Toeplitz operators and Toeplitz matrices the following results can be formulated both for Toeplitz operators and Toeplitz matrices.

Proposition A.13

If $a\in C(S^1)$ and $\{a(z),z\in S^1\}$ is a line segment, then $\mathrm{spec} {\mathscr {T}}(a)=\{a(z),z\in S^1\}$.

Proof

Follows from [5, Theorem 1.17]. $\square$

In the following we consider matrix valued operators.

Definition A.14

(Block operator) For $a\in [L^{\infty }(S^1)]^{2\times 2}$ we define the block multiplication operator ${\mathscr {Q}}(a)\in \mathscr {B}([L^2(S^1)]^{2})$ by $\left( {\mathscr {Q}}(a)u\right) (z):=a(z)\, u(z)$ for $u \in [L^2(S^1)]^{2}$ and $z \in S^1$.

Using the orthogonal projection $\mathscr {P}:[L^2(S^1)]^{2}\rightarrow [H^+(S^1)]^{2}$, we define the block Toeplitz operator ${\mathscr {T}}(a)\in \mathscr {B}([H^+(S^1)]^{2})$ by ${\mathscr {T}}(a) u:=\mathscr {P}{\mathscr {Q}}(a)u$ for $u\in [H^+(S^1)]^{2}$.

Definition A.15

(Block Toeplitz matrix) An infinite matrix $T\in \mathscr {B}(l^2(\mathbb {N}_0))$ with $(Tf)_n:=\sum _{m=0}^\infty T_{n,m}f_m$ is called an (infinite) $2\times 2$ block Toeplitz matrix if $T_{n,m}= T_{n+2,m+2}$ for all $n,m\in \mathbb {N}_0$. For the block Toeplitz operator ${\mathscr {T}}(a):[H^+(S^1)]^2 \rightarrow [H^+(S^1)]^2$ with symbol $a\in [L^{\infty }(S^1)]^{2\times 2}$ the associated infinite $2\times 2$ block Toeplitz matrix $T(a):l^2(\mathbb {N}_0)\rightarrow l^2(\mathbb {N}_0)$ is given by

$$\begin{aligned} \left( {\begin{matrix} T(a)_{2n,2m} &{}\,\, T(a)_{2n,2m+1}\\ T(a)_{2n+1 ,2m} &{}\,\, T(a)_{2n+1,2m+1} \end{matrix}} \right) :=\int _{S^1} z^{n-m} a(z) |dz|,\qquad n,m \in \mathbb {N}_0. \end{aligned}$$

(A.5)

Note that T(a) is the matrix representation of ${\mathscr {T}}(a)$ with respect to the orthonormal basis $\Big \{\left( {\begin{matrix} z^0\\ 0 \end{matrix}} \right) , \left( {\begin{matrix} 0\\ z^0 \end{matrix}} \right) ,$ $\left( {\begin{matrix} z^1\\ 0 \end{matrix}} \right) , \left( {\begin{matrix} 0\\ z^1 \end{matrix}} \right) , \dots \Big \}$ of $[H^+(S^1)]^2$. If T is an infinite $2\times 2$ block Toeplitz matrix, there exists $2\times 2$ matrices $a_k\in \mathbb {C}^{2\times 2}$ for $k\in \mathbb {Z}$ such that (A.5) holds true with the right hand side replaced by $a_{n-m}$. Then $T=T(a)$ can be shown to be associated to the block Toeplitz operator ${\mathscr {T}}(a)$ with symbol $a(z):=\sum _{k\in \mathbb {Z}}z^k a_k$ for $z \in S^1$. Hence, there is again a one-to-one correspondence of block Toeplitz operators and block Toeplitz matrices.

Theorem A.16

If $a\in [C(S^1)]^{2\times 2}$ and a(z) is Hermitian positive definite for all $z\in S^1$, then $\mathrm{spec} {\mathscr {T}}(a)$ is contained in the convex hull of $S:=\bigcup _{z\in S^1}\{speca(z)\}$.

Proof

This follows from the Lemma of Lax-Milgram: If $\lambda >\sup (S)$, then

$$\begin{aligned}&\left\langle u, (\lambda \mathrm id-{\mathscr {T}}(a))u \right\rangle _{[H^+(S^1)]^2} = \int _{S^1} u(z)^H(\lambda \mathrm id-a(z))u(z) |dz|\\&\quad \ge (\lambda -\sup (S))\Vert u\Vert _{[L^2(S^2)]^2}^2 \end{aligned}$$

so $\lambda \notin \mathrm{spec} {\mathscr {T}}(a)$. Here $v^H$ denotes the Hermitian transpose of a vector v. The cases $\lambda <\inf (S)$ and $\lambda \in \mathbb {C}{{\setminus }}\mathbb {R}$ can be treated similarly. $\square$