1. Introduction and motivations
Let
$\mathbb {D}$
be the open unit disc in the complex plane
$\mathbb {C}$
, and
$L^2$
and
$L^\infty $
the Lebesgue spaces of square integrable and essentially bounded measurable complex functions, respectively, on the circle
$\partial \mathbb {D}$
equipped with the normalized linear Lebesgue measure
$d\theta $
. Let
$H^2\subset L^2$
be the Hardy subspace, and
$H^\infty \subset L^\infty $
the subalgebra of boundary functions of bounded analytic functions in
$\mathbb {D}$
. Let C and
$QC:=(H^\infty +C)\bigcap \overline {H^\infty +C}=VMO\bigcap L^\infty $
[Reference Sarason14] be the C*-subalgebras of
$(L^\infty ,\|\cdot \|_\infty )$
consisting, respectively, of continuous and quasicontinuous functions on
$\partial \mathbb {D}$
. Here,
$VMO$
is the space of
$L^1$
functions on
$\partial \mathbb {D}$
of vanishing mean oscillation [Reference Sarason14].
For
$f\in L^\infty $
,
$M_f\in \mathcal {L}(L^2)$
denotes the multiplication operator by f on
$L^2$
whose compression to
$H^2$
,
$T_f:=PM_f|H^2,$
is the Toeplitz operator with symbol f. Here,
$P\in \mathcal {L}(L^2)$
is the orthogonal projection onto
$H^2$
. Let
$C_{\bar {z}}\in \mathcal {L}(L^2)$
be the composition operator by the complex conjugate
$z\mapsto \bar {z}$
on
$\partial \mathbb {D}$
and write
$\tilde {f}:=C_{\bar {z}}f=f\circ \bar {z}$
for
$f\in L^2$
. Define the Hankel operator
$H_f$
with symbol
$f\in L^\infty $
to be the compression of
$C_{\bar {z}}M_zM_f\in \mathcal {L}(L^2)$
to
$H^2$
$$ \begin{align*}H_f:=PC_{\bar{z}}M_zM_f|H^2=C_{\bar{z}}M_z(I-P)M_f|H^2.\end{align*} $$
See [Reference Böttcher and Silbermann3] for basic properties. The following identity is well known and useful:
$$ \begin{align}H_{fg}=H_f T_g + T_{\tilde{f}} H_g, \quad f,g\in L^\infty. \end{align} $$
For a subset
$\Omega \subset L^\infty $
,
$\mathcal {T}(\Omega )$
denotes the norm-closed subalgebra of
$\mathcal {L}(H^2)$
generated by the Toeplitz operators with symbol class
$\Omega $
, and
$\mathcal {K}$
the ideal of compact operators on
$H^2$
. The essential commutant of
$\mathcal {S}\subset \mathcal {L}(H^2)$
is defined as
$\mathcal {S}^{\prime }_e=\{T\in \mathcal {L}(H^2):TS-ST\in \mathcal {K},\forall S\in \mathcal {S}\}$
. Barría and Halmos [Reference Barría and Halmos2] asked whether Hankel operators in
$\{T_z\}^{\prime }_e=\mathcal {T}(C)^{\prime }_e$
must lie in the Toeplitz algebra
$\mathcal {T}(L^\infty )$
. The answer turned out to be negative [Reference Chen and Chen4, Theorem 4.6], and several constructions [Reference Barría1, Reference Chen, Guo, Izuchi and Zheng5, Reference Yan17] of
$f\in QC$
and Blaschke products b are known such that the commutator
$[T_z,H_{\bar b}]\in \mathcal {K}$
, but
$[T_f,H_{\bar b}]\notin \mathcal {K}$
, so that
$H_{\bar b}\notin \mathcal {T}(QC)^{\prime }_e\supset \mathcal {T}(L^\infty )$
. A naturally revised question would then ask if Hankel operators in
$\mathcal {T}(QC)^{\prime }_e$
must be in the Toeplitz algebra. This question is of interest for although
$\mathcal {T}(L^\infty )^{\prime }_e=\mathcal {T}(QC)$
[Reference Davidson6],
$\mathcal {T}(QC)^{\prime }_e\supsetneq \mathcal {T}(L^\infty )$
[Reference Xia16]. Thus, it is natural to seek concrete classes
$\mathcal {S}$
of operators satisfying
$\mathcal {T}(QC)^{\prime }_e\bigcap \mathcal {S}=\mathcal {T}(L^\infty )\bigcap \mathcal {S}$
.
There is an equivalent formulation in terms of Douglas algebras, that is, closed subalgebras of
$L^\infty $
containing
$H^\infty $
. Define
$$ \begin{align*}B_T&=\{f\in L^\infty: H_f\in \mathcal{T}(L^\infty)\}, \\ B_{QC}&=\{f\in L^\infty: H_f\in \mathcal{T}(QC)^{\prime}_e\}, \\ B_{C}&=\{f\in L^\infty: H_f\in \mathcal{T}(C)^{\prime}_e\}. \end{align*} $$
It follows from standard properties of Toeplitz and Hankel operators that these are all Douglas algebras, where in particular closure under products is due to (1.1). Then, in view of the preceding discussions and the construction in [Reference Chen, Guo, Izuchi and Zheng5, Section 3], one has the following observation.
Proposition 1.1 The Douglas algebras satisfy
$H^\infty +C\subsetneq B_T \subset B_{QC}\subsetneq B_C \subsetneq L^\infty $
.
The revised Barría–Halmos question asks if indeed
$B_T=B_{QC}$
?
While it is difficult to determine
$B_T$
, using the Chang–Marshall structure theorem (cf. [Reference Garnett7]) for Douglas algebras, we shall characterize
$B_{QC}$
. Note that [Reference Chen and Chen4, Proposition 3.5] (also see [Reference Barría1, Proposition, p. 1508]) has characterized the Blaschke products invertible in
$B_C$
. Our main results are:
Theorem 1.2 If b is an interpolating Blaschke product, then
$\bar b\in B_{QC}$
if and only if the zero sequence
$\{z_n\}_n$
of b in
$\mathbb {D}$
lies in the union of a Stolz angle at
$\lambda =1$
and another at
$\lambda =-1$
.
Theorem 1.3 The set
$B_{QC}$
does not contain the complex conjugate of any nonconstant singular inner function.
In other words, the inner functions invertible in
$B_{QC}$
are all Blaschke products. Incidentally, there follows a class of atomic singular inner functions s such that
$H_{\bar s}\in \mathcal {T}(C)^{\prime }_e\backslash \mathcal {T}(QC)^{\prime }_e$
, adding to the Blaschke-product examples in [Reference Barría1, Reference Chen, Guo, Izuchi and Zheng5, Reference Yan17].
2. The Douglas algebra
$B_{QC}$
The maximal ideal space
$M(A)$
of a commutative unital Banach algebra A is equipped with the Gelfand topology, and the Gelfand transform of
$a\in A$
on
$M(A)$
is denoted by the same letter. The fiber
$M_x(A)$
consists of the extensions in
$M(A)$
of
$x\in M(U)$
of a closed subalgebra U. Douglas algebras B are uniform algebras on
$M(L^\infty )$
, and
$M(B)$
is homeomorphically identified with its trace in
$M(H^\infty )$
(cf. [Reference Garnett7]). In particular,
$M(H^\infty +C)=M(H^\infty )\backslash \mathbb {D}$
under the natural embedding
$\mathbb {D}\hookrightarrow M(H^\infty )$
. The support set of the unique representing measure
$\mu _m$
on
$M(L^\infty )$
for
$m\in M(H^\infty )$
is denoted by
$S_m$
.
The proof of the following lemma involves only obvious changes to that of [Reference Gorkin and Zheng9, Lemma 1.5], and is omitted.
Lemma 2.1 For the Douglas algebra
$H^\infty [\Omega ]$
generated by
$\Omega \subset L^\infty $
over
$H^\infty $
,
$$ \begin{align*}M(H^\infty[\Omega])=\{m\in M(H^\infty): f|S_m\in H^\infty|S_m,\forall f\in\Omega\}.\end{align*} $$
Recall that
$m\in M(H^\infty +C)$
is a trivial point if its Gleason part is a singleton (cf. [Reference Garnett7]). Let
$\mathcal {G}$
be the set of nontrivial points of
$M(H^\infty +C)$
. It is well known that
$\mathcal {G}\subsetneq M(H^\infty +C)\backslash M(L^\infty )$
. For
$h\in H^\infty $
, put
$$ \begin{align*}Z(h)&:=\{m\in M(H^\infty+C): h(m)=0\}, \\ Z(h)|QC&:=\{m|QC: m\in Z(h)\}\subset M(QC), \\ (Z(h)\bigcap\mathcal{G})|QC&:=\{m|QC: m\in Z(h)\bigcap\mathcal{G}\}. \end{align*} $$
Some results on
$M(QC)\subset QC^*$
are needed. For
$\delta>0$
, let
$m_{\lambda ,\delta }\in QC^*$
be the averaging functional over the subarc
$(\lambda -\delta ,\lambda +\delta )$
of
$\partial \mathbb {D}$
centered at
$\lambda $
and with
$\theta $
-measure
$2\delta $
. Let
$v_z\in QC^*$
be the harmonic-extension functional at
$z\in \mathbb {D}$
. Sarason [Reference Sarason15, pp. 822–823] defined and showed that in the weak-star topology of
$QC^*$
,
$$ \begin{align}M_\lambda^0(QC)&:=\overline{\{m_{\lambda,\delta}:\delta>0\}}\bigcap M(QC)=\{\lim_\omega m_{\lambda,\delta_\omega}: \ \lim_\omega \delta_\omega=0\} \end{align} $$
$$ \begin{align} \phantom{M_\lambda^0(QC)\;\;}&=\overline{\{v_z: z\in A_\lambda\}}\bigcap M(QC)=\{\lim_\omega v_{z_\omega}: \ \lim_\omega z_\omega=\lambda, \ z_\omega\in A_\lambda\}, \end{align} $$
where, for
$\rho \in [1,\infty )$
,
$$ \begin{align*}A_\lambda=A_{\lambda,\rho}:=\{z\in\mathbb{D}:|z-\lambda|\le\rho(1-|z|)\}\end{align*} $$
is any Stolz angle at
$\lambda $
. Note
$M_\lambda ^0(QC)$
is a small subset of the fiber
$M_\lambda (QC)$
over
$\lambda \in \partial \mathbb {D}\cong M(C)$
. Because
$QC=VMO\bigcap L^\infty $
is invariant for the operator
$C_{\bar {z}}$
,
$M(QC)\subset QC^*$
is invariant for the adjoint
$C_{\bar {z}}^*\in \mathcal {L}(QC^*)$
. Writing
$\bar {x}:=C_{\bar {z}}^* x=x\circ C_{\bar {z}}\in M(QC)$
for
$x\in M(QC)$
, one has the useful fact [Reference Power13] that
$\bar {x}=x$
if and only if
$$ \begin{align*}x\in M_1^0(QC)\bigsqcup M_{-1}^0(QC)=:K^0.\end{align*} $$
The next result identifies the inner functions invertible in
$B_{QC}$
.
Proposition 2.2 For an inner function
$u\in H^\infty $
, the following are equivalent:
-
(1)
$\bar u\in B_{QC}$
. -
(2)
$Z(u)|QC\subset K^0$
. -
(3)
$(Z(u)\bigcap \mathcal {G})|QC\subset K^0$
.
Proof By (1.1) and Hartman’s theorem (cf. [Reference Böttcher and Silbermann3]),
$[T_f,H_{\bar u}]-H_{(\tilde f -f)\bar u}\in \mathcal {K}$
for every
$f\in QC$
. So (1) is equivalent to
$(\tilde f -f)\bar u\in H^\infty +C$
for every
$f\in QC$
, which is in turn equivalent to
$H^\infty [(\tilde f -f)\bar u: f\in QC]\subset H^\infty +C$
, and to
$$ \begin{align}M(H^\infty+C)\subset M(H^\infty[(\tilde f -f)\bar u: f\in QC]) \end{align} $$
by the Chang–Marshall theorem. In view of Lemma 2.1, (2.3) amounts to
$$ \begin{align}(\tilde f -f)\bar u|S_m\in H^\infty|S_m, \quad \forall f\in QC, \ \forall m\in M(H^\infty+C). \end{align} $$
Notice for
$x\in M(QC)$
that
$m\in M_x(H^\infty +C)\Longleftrightarrow S_m\subset M_x(L^\infty )$
, so that
$$ \begin{align*}(\tilde f -f)|S_m\equiv x(\tilde f)-x(f)=f(\bar x)-f(x),\quad f\in QC.\end{align*} $$
Therefore,
$(\tilde f -f)\bar u|S_m\equiv 0$
for all
$m\in M_x(H^\infty +C)$
,
$x\in K^0$
. Then, because, for every
$x\in M(QC)\backslash K^0$
,
$f(\bar x)-f(x)\ne 0$
for some
$f\in QC$
, while the fiber
$M_x(H^\infty +C)$
is nonempty for the C*-subalgebra
$QC$
, (2.4) is seen equivalent to
$$ \begin{align}\bar u|S_m\in H^\infty|S_m, \quad \forall m\in M_x(H^\infty+C),\ x\in M(QC)\backslash K^0. \end{align} $$
It remains to show (2.5)
$\Longrightarrow $
(2), and (3)
$\Longrightarrow $
(2.5) using some argument similar to that in the proof of [Reference Yan17, Theorem 4.4] based on [Reference Gorkin and Mortini8].
That (2.5)
$\Longrightarrow $
(2) is due to
$\bar u|S_m\in H^\infty |S_m\Longleftrightarrow u|S_m\equiv c\Longleftrightarrow u(m)=c$
for a unimodular constant c. Next, assume (3) and let
$x\in M(QC)\backslash K^0$
,
$m\in M_x(H^\infty +C)$
be arbitrary. If
$u|S_m$
is not invertible in the restriction algebra
$H^\infty |S_m$
, then, by the identification of
$M(H^\infty |S_m)$
(cf. [Reference Garnett7]),
$u(\xi )=0$
for some
$\xi \in M(H^\infty +C)$
with
$S_{\xi }\subset S_m$
,
$S_{\xi }$
not a singleton for
$|u|\equiv 1$
on
$M(L^\infty )$
. By [Reference Gorkin and Mortini8, Corollary 3.2],
$\xi \in M(H^\infty +C)\backslash M(L^\infty )$
lies in the closure in
$M(H^\infty +C)$
of the subset
$\{y\in \mathcal {G}: S_y\subset S_\xi \}$
. Thus, for each
$n\in \mathbb {N}$
, there exists
$y_n\in \mathcal {G}$
with
$S_{y_n}\subset S_\xi $
and
$|u(y_n)|<1/n$
. By [Reference Gorkin and Mortini8, Theorem 1.2], the sequence
$\{y_n\}_n$
in
$\mathcal {G}$
has a cluster point
$y\in \mathcal {G}$
. We have
$S_y\subset S_\xi \subset S_m$
due to each
$S_{y_n}\subset S_\xi $
, and
$u(y)=0$
for
$u(y_n)\rightarrow 0$
. Now, it follows that
$$ \begin{align*}x=y|QC\in(Z(u)\bigcap\mathcal{G})|QC,\end{align*} $$
which is a contradiction to (3). Therefore,
$u|S_m$
is invertible in
$H^\infty |S_m$
, that is,
$\bar {u}|S_m\in H^\infty |S_m$
. So, (2.5) holds, and the proof is complete. ▪
Remark 2.3 In lieu of the Chang–Marshall theorem, the proof could be based on the Bishop antisymmetry theorem (cf. [Reference Garnett7]), or a characterization of essentially commuting Hankel and Toeplitz operators with general symbols [Reference Guo and Zheng11, Theorem 2].
In addition to specializations of Proposition 2.2, the proofs of Theorems 1.2 and 1.3 will rely on a technical result involving the construction of certain
$QC$
functions. The next proof closely follows [Reference Yan17, Example 1]. Also see [Reference Barría1, Reference Xia16].
Lemma 2.4 If
$z_n\in \mathbb {D}$
,
$\lambda \in \partial \mathbb {D}$
, and
$h\in H^\infty $
are such that the sequence
$\{z_n\}_n$
is not contained in any Stolz angle at
$\lambda $
while
$z_n\rightarrow \lambda $
and
$h(z_n)\rightarrow 0$
, then
Proof After taking a rotation, we can and do assume
$\lambda =1$
. For each
$k\in \mathbb {N}$
, put
$$ \begin{align*}\rho_k=4\pi\exp(2^k)+1.\end{align*} $$
Because
$\{z_n\}_n$
is not contained in any Stolz angle
$A_1$
and
$z_n\rightarrow 1$
, one can inductively choose an increasing index sequence
$\{n_k: k\in \mathbb {N}\}$
such that
$r_k:=|\arg z_{n_k}|/(2\pi )$
with
$\arg \in (-\pi , \pi ]$
satisfies
$0<r_{k+1}<r_k/2$
, that
$t_k:=\exp (2^k)(1-|z_{n_k}|)$
satisfies
$t_{k+1}<t_k$
, and that
$z_{n_k}\notin A_{1,\rho _k}$
. By elementary geometry, the latter implies
$$ \begin{align*}2\pi r_k+(1-|z_{n_k}|)>|z_{n_k}-1|>\rho_k(1-|z_{n_k}|),\end{align*} $$
so that
$r_k>2\exp (2^k)(1-|z_{n_k}|)=2t_k$
.
Let
$J_k\subset I_k$
be a pair of concentric subarcs of
$\partial \mathbb {D}$
centered at
$z_{n_k}/|z_{n_k}|$
with
$\theta (J_k)=1-|z_{n_k}|$
and
$\theta (I_k)=t_k$
. Let
$f_k\in C$
be
$[0,1]$
-valued with
$f_k\equiv 1$
on
$J_k$
,
$f_k\equiv 0$
off
$I_k$
, and the maximum-mean-oscillation norm
$$ \begin{align*}\|f_k\|_{BMO}\le M/\ln[\theta(I_k)/\theta(J_k)]=M2^{-k}\end{align*} $$
for a universal constant M [Reference Guillory and Sarason10, Lemma 1]. Notice that because
$$ \begin{align*}r_k-r_{k+1}>r_k/2>t_k>(t_k+t_{k+1})/2,\end{align*} $$
the support subarcs
$I_k$
are pairwise disjoint. Now, assuming
$\arg z_{n_k}>0$
for infinitely many k, define the pointwise convergent series of continuous functions
$$ \begin{align*}\sum_{\arg z_{n_k}>0} f_k \ =\phi\in VMO\bigcap L^\infty=QC \end{align*} $$
due to
$C\subset VMO$
and
$$ \begin{align*}\sum_{\arg z_{n_k}>0} \|f_k\|_{BMO}\le\sum \|f_k\|_{BMO}<\infty.\end{align*} $$
The function
$\phi $
vanishes on one side of
$\lambda =1$
, so the integral gap
$\gamma _1(\phi )=0$
at
$1$
[Reference Sarason15] together with (2.1) implies [Reference Yan17]
$$ \begin{align}\phi|M_1^0(QC)\equiv 0. \end{align} $$
Also, it follows from construction and [Reference Sarason15, Lemma 5] that the harmonic extension
$\phi (z_{n_k})\rightarrow 1$
for
$\arg z_{n_k}>0$
and
$k\rightarrow \infty $
. Thus, if
$m\in M(H^\infty )$
is a cluster point of these
$z_{n_k}\in \mathbb {D}\hookrightarrow M(H^\infty )$
, then
$m\in M_1(H^\infty +C)$
for
$z_{n_k}\rightarrow 1$
, while (cf. [Reference Garnett7])
$$ \begin{align}\phi(m|QC)=\int_{M(L^\infty)}\phi\ d\mu_m=\lim_\omega \int_{M(L^\infty)}\phi\ d\mu_{z_{n_{k_\omega}}}=\lim_\omega \phi(z_{n_{k_\omega}})=1 \end{align} $$
for a subnet of the sequence
$\{z_{n_k}: k\in \mathbb {N},\ \arg z_{n_k}>0\}$
converging to m. It follows from (2.6) and (2.7) that
$m|QC\in M_1(QC)\backslash M_1^0(QC)$
, while
$h(m)=0$
due to the hypothesis
$h(z_n)\rightarrow 0$
. The existence of such
$m\in Z(h)$
completes the proof. ▪
We are ready to prove Theorem 1.2.
Proof If the zero sequence
$\{z_n\}_n$
in
$\mathbb {D}$
of an interpolating Blaschke product b lies in
$A_1\bigcup A_{-1}$
for Stolz angles
$A_{\pm 1}$
at
$\pm 1$
, then, by [Reference Hoffman12, p. 206, (v)],
$$ \begin{align}Z(b)=\overline{\{z_n\}_n}\backslash \mathbb{D}\subset (\overline{A_1}\backslash \mathbb{D}) \bigcup (\overline{A_{-1}}\backslash \mathbb{D}), \end{align} $$
where closures are taken in
$M(H^\infty )$
. In view of the homeomorphic embedding
$m\in M(H^\infty )\mapsto \mu _m\in (C(M(L^\infty )))^*$
in the weak-star topology, one has
$$ \begin{align*}\overline{A_{\pm 1}}\backslash \mathbb{D}\cong\overline{\{\mu_z: z\in A_{\pm 1}\}} \backslash \{\mu_z:z\in\mathbb{D}\}\subset \{\mu_m: m\in M(H^\infty+C)\}. \end{align*} $$
By continuity of the restrictions
$(C(M(L^\infty )))^*\rightarrow (H^\infty +C)^*\rightarrow QC^*$
, and because
$\mu _z$
corresponds to the harmonic-extension functional on
$L^\infty $
, one further deduces
$$ \begin{align} (\overline{A_{\pm 1}}\backslash \mathbb{D})|QC\subset\overline{\{v_z: z\in A_{\pm 1}\}} \bigcap M(QC), \end{align} $$
where the latter closure is taken in the weak-star topology of
$QC^*$
. Now, it follows from (2.2), (2.8), and (2.9) that
$Z(b)|QC\subset K^0$
, and
$\bar b\in B_{QC}$
by Proposition 2.2.
Next, suppose
$\{z_n\}_n$
does not lie in the union of two Stolz angles at
$\pm 1$
, respectively, and we shall derive
$\bar b\notin B_{QC}$
. If
with closure in
$\overline {\mathbb {D}}$
, then, by [1, Reference Chen and Chen4],
$\bar b\notin B_C\supset B_{QC}$
. It remains to consider the other case
$$ \begin{align}\overline{\{z_n\}_n}\bigcap\partial\mathbb{D}\subset \{\pm 1\}. \end{align} $$
By hypothesis, for every
$k\in \mathbb {N}$
, there exists
$z_{n_k}\notin A_{1,k}\bigcup A_{-1,k}$
. Passing to a subsequence of
$\{z_{n_k}\}_k$
if needed, we can take
$\lim _k z_{n_k}=\lambda \in \overline {\mathbb {D}}$
. Noting
$$ \begin{align*}z_{n_k}\notin A_{1,k}\supset\{z\in\mathbb{D}: |z|\le(k-1)/(k+1)\},\end{align*} $$
one has
$|\lambda |=\lim _k |z_{n_k}|=1$
; hence,
$\lambda =1$
or
$-1$
by (2.10). Because the sequence
$\{z_{n_k}\}_k$
does not lie in any Stolz angle at
$\lambda $
for
$z_{n_k}\notin A_{\lambda ,k}$
, Lemma 2.4 yields
. So, Proposition 2.2(2) fails and
$\bar b\notin B_{QC}$
.▪
The following characterization of
$B_{QC}$
is an immediate consequence of Proposition 1.1, (the full strength of) the Chang–Marshall theorem, and Theorem 1.2.
Corollary 2.5 If
$f\in L^\infty $
, then
$H_f\in \mathcal {T}(QC)^{\prime }_e$
if and only if f lies in the closed subalgebra of
$L^\infty $
generated by
$H^\infty $
and the complex conjugates of all interpolating Blaschke products whose zero sequence in
$\mathbb {D}$
is contained in the union of a Stolz angle at
$1$
and another at
$-1$
.
Now, we prove Theorem 1.3.
Proof Suppose
$\bar s\in B_{QC}$
for a singular inner function s whose finite positive singular measure
$d\mu \perp d\theta $
on
$\partial \mathbb {D}$
has support set
$\text {supp}\mu $
. For every
$\lambda \in \text {supp}\mu $
, there exists a sequence
$z_n\in \mathbb {D}$
such that
$z_n\rightarrow \lambda $
while
$s(z_n)\rightarrow 0$
[Reference Hoffman12, p. 76, Exercise 13], [Reference Böttcher and Silbermann3, p. 31]. Let
$m\in M(H^\infty )$
be a cluster point of the sequence
$z_n\in \mathbb {D}\hookrightarrow M(H^\infty )$
. Clearly,
$m\in M_\lambda (H^\infty +C)$
while
$s(m)=0$
. By Proposition 2.2,
$m|QC\in Z(s)|QC\subset K^0$
; thus,
$\lambda =m|C\in \{\pm 1\}$
. That is,
$\text {supp}\mu \subset \{\pm 1\}$
, so that for
$z\in \mathbb {D}$
the integral representation of s admits the form [Reference Hoffman12]
$$ \begin{align*}s(z)=e^{it}\exp\left(-\int_{\{\pm 1\}} \frac{\lambda+z}{\lambda-z}d\mu(\lambda)\right)&=e^{it}\exp\left(-r^+\frac{1+z}{1-z}\right)\exp\left(r^-\frac{z-1}{z+1}\right) \\ &=:e^{it}s_1(z) s_{-1}(z), \end{align*} $$
where
$r^{\pm }=\mu (\pm 1)\ge 0$
. Note that
$\bar s_1=e^{it}s_{-1}\bar s\ \theta $
-a.e. on
$\partial \mathbb {D}$
, so that
$\bar s_1\in B_{QC}$
.
Let
$t_n\in (0,1)$
be a sequence such that
$t_n\uparrow 1$
. For each n, let
$A_{1,n}$
be the Stolz angle at
$1$
, and
$O_n$
the circle with center
$t_n$
and radius
$1-t_n$
. Noting that the Cayley transform
$z\mapsto {(1+z)}/{(1-z)}$
conformally maps
$O_n\backslash \{1\}$
onto the vertical line
$\{w\in \mathbb {C}:\Re w=t_n/(1-t_n)\}$
, one has
$$ \begin{align}|s_1(z)|\equiv \exp(-r^+ t_n/(1-t_n)),\quad z\in O_n\backslash\{1\}\subset\mathbb{D}. \end{align} $$
Because the inner circle
$O_n$
is tangent to the outer circle
$\partial \mathbb {D}$
at
$1$
, one can find by elementary geometry
$z_n\in O_n\backslash \{1\}$
such that
$|z_n-1|<1/n$
and
$z_n\notin A_{1,n}$
.
By construction, the sequence
$\{z_n\}_n$
in
$\mathbb {D}$
is not contained in any Stolz angle at
$1$
, while
$z_n\rightarrow 1$
and
$s_1(z_n)\rightarrow 0$
by (2.11) unless
$r^+=0$
. If
$r^+>0$
, Lemma 2.4 implies
, but
$\bar s_1\in B_{QC}$
and Proposition 2.2 imply otherwise, a contradiction. So,
$r^+=0$
. The same argument applies to the other factor
$s_{-1}$
to give
$r^-=0$
as well, and
$s\equiv e^{it}$
a constant as required. ▪
Remark 2.6 If
$r>0$
and
$s(z)=\exp (r {(z+1)}/{(z-1)})$
on
$\mathbb {D}$
, then
$H_{\bar s}\in \mathcal {T}(C)^{\prime }_e\backslash \mathcal {T}(QC)^{\prime }_e$
. For,
$\bar s\notin B_{QC}$
by Theorem 1.3, while the boundary function on
$\partial \mathbb {D}$
of the analytic function
$s(z)$
in
$z\in \mathbb {C}\backslash \{1\}$
is of course continuous at every point of
$\partial \mathbb {D}\backslash \{1\}$
. Hence,
$(\bar z -z)\bar s\in C$
and
$H_{\bar s}\in \mathcal {T}(C)^{\prime }_e$
.
This section ends with some observations and an open question about
$Z(s)$
of
$s(z)=\exp (r {(z+1)}/{(z-1)})$
. Because continuity of
$s\in L^\infty $
at every
$\lambda \in \partial \mathbb {D}\backslash \{1\}$
is equivalent to
$s|M_\lambda (L^\infty )$
being a unimodular constant (cf. [Reference Böttcher and Silbermann3, Prop. 2.79(b)]), and because
$m\in M_\lambda (H^\infty +C)\Longleftrightarrow S_m\subset M_\lambda (L^\infty )$
, one has
$Z(s)|QC\subset M_1(QC)$
. By (2.2) with
$A_{1,1}=[0,1)$
, every
$x\in M_1^0(QC)$
is the limit in
$QC^*$
of a net
$\{v_{t_\omega }\}_\omega $
,
$t_\omega \in [0,1)$
and
$\lim _\omega t_\omega =1$
. Passing to a subnet, one has
$\lim _\omega t_\omega =m\in M(H^\infty +C)$
in
$M(H^\infty )$
. It follows that
$s(m)=\lim _\omega s(t_\omega )=0$
and
$m|QC=\lim _\omega v_{t_\omega }=x$
in
$QC^*$
. That is,
$M_1^0(QC)\subset Z(s)|QC$
, while the proof of Theorem 1.3 shows the inclusion proper. In summary,
$$ \begin{align*}M_1^0(QC)\subsetneq Z(s)|QC\subset M_1(QC).\end{align*} $$
Is it indeed that
$Z(s)|QC=M_1(QC)$
?
Acknowledgment
I would like to express my sincere gratitude for Professor Tyrone Duncan and Professor Albert Sheu for guidance and encouragement, and the reviewer for helpful comments and suggestions.
