New York Journal of Mathematics

New York J. Math. 14 (2008) 431–457.

N_{φ}-type quotient modules on the torus

Keiji Izuchi and Rongwei Yang

Abstract. Structure of the quotient modules in H^{2}(Γ^{2}) is very complicated. A good understanding of some special examples will shed light on the general picture. This paper studies the so-called N_{φ}-type quotient modules, namely, quotient modules of the form H^{2}(Γ^{2}) ⊖ [z − φ], where φ(w) is a function in the classical Hardy space H^{2}(Γ) and [z − φ] is the submodule generated by z − φ(w). This type of quotient module provides good examples in many studies. A notable fact is its close connections with some classical operators, namely the Jordan block and the Bergman shift. This paper studies spectral properties of the compressions S_{z} and S_{w}, compactness of evaluation operators, and essential reductivity of H^{2}(Γ^{2}) ⊖ [z − φ].

Contents

1. Introduction 431 2. Preliminaries 433 3. The spectra of S_{z} and S_{w} 436 4. Compactness of L(0)|_{N_{φ}} and D_{z} 443 5. The case when φ is inner 450 References 456

1. Introduction

Let H^{2}(Γ^{2}) be the Hardy space on the two-dimensional torus Γ^{2}. We denote by z and w the coordinate functions. Shift operators T_{z} and T_{w} on H^{2}(Γ^{2}) are defined by T_{z}f = zf and T_{w}f = wf for f ∈ H^{2}(Γ^{2}). Clearly, both T_{z} and T_{w} have infinite multiplicity. A closed subspace M of H^{2}(Γ^{2})

Received December 12, 2007. Mathematics Subject Classification. Primary 46E20; Secondary 47A13. Key words and phrases. The Hardy space on the torus, quotient modules, two variable

Jordan block, evaluation operators, essential reductivity.

The first author is partially supported by Grant-in-Aid for Scientific Research (No.16340037), Ministry of Education, Science and Culture.

431^{ISSN 1076-9803/08}

is called a submodule (over the algebra H^{∞}(𝔻^{2})), if it is invariant under multiplications by functions in H^{∞}(𝔻^{2}). Here 𝔻 stands for the open unit disk. Equivalently, M is a submodule if it is invariant for both T_{z} and T_{w}. The quotient space N := H^{2}(Γ^{2}) ⊖ M is called a quotient module. Clearly T_{z}^{∗}N ⊂ N and T_{w}^{∗}N ⊂ N. And for this reason N is also said to be backward shift invariant. In the study here, it is necessary to distinguish the classical Hardy space in the variable z and that in the variable w, for which we denote by H^{2}(Γ_{z}) and H^{2}(Γ_{w}), respectively. H^{2}(Γ_{z}) and H^{2}(Γ_{w}) are thus different subspaces in H^{2}(Γ^{2}). We will simply write H^{2}(Γ) when there is no need to tell the difference. In H^{2}(Γ), it is well-known as the Beurling theorem that if M ⊂ H^{2}(Γ) is invariant for T_{z}, then M = qH^{2}(Γ) for an inner function q(z). The structure of submodules in H^{2}(Γ^{2}) is much more complex, and there has been a great amount of work on this subject in recent years. A good reference of this work can be found in [3]. One natural approach to the problem is to find and study some relatively simple submodules, and hope that the study will generate concepts and general techniques that will lead to a better understanding of the general picture. This in fact has become an interesting and encouraging work.

In this paper, we look at submodules of the form [z − φ(w)], where φ is a function in H^{2}(Γ_{w}) with φ ≠ 0 and [z−φ(w)] is the closure of (z−φ)H^{∞}(Γ^{2}) in H^{2}(Γ^{2}). For simplicity we denote [z − φ(w)] by M_{φ}. One good way of studying M_{φ} is through the so-called two variable Jordan block (S_{z}, S_{w}) defined on the quotient module

N_{φ} := H^{2}(Γ^{2}) ⊖ M_{φ}.

For every quotient module N, the two variable Jordan block (S_{z}, S_{w}) is the compression of the pair (T_{z}, T_{w}) to N, or more precisely,

S_{z}f = P_{N}zf, S_{w}f = P_{N}wf, f ∈ N,

where P_{N} : H^{2}(Γ^{2}) → N is the orthogonal projection. This paper studies interconnections between the quotient module N_{φ}, the two variable Jordan block (S_{z}, S_{w}) and the function φ. Some related work has been done in [14, 22, 23]. By [14], N_{φ} ≠ {0} if and only if φ(𝔻) ∩ 𝔻 ≠ ∅. If φ = 0, then M_{φ} = zH^{2}(Γ^{2}) and N_{φ} = H^{2}(Γ_{w}), so we assume that φ ≠ 0. For convenience, we let

Ω_{φ} = {w ∈ 𝔻 : |φ(w)| < 1},

and assume throughout the paper that N_{φ} ≠ {0}, i.e., φ(𝔻) ∩ 𝔻 ≠ ∅. The paper is organized as follows.

Section 1 is the introduction.

Section 2 introduces some useful tools and states a few related known results.

Section 3 studies the spectral properties of the operators S_{z} and S_{w}. It is interesting to see how these properties depend on the function φ.

N_{φ}-type quotient modules on the torus 433

A notable phenomenon in many cases is the compactness of the defect operators I −S_{z}S_{z}^{∗} and I −S_{z}^{∗}S_{z}. Section 4 aims to study how the compactness is related to the properties of φ.

The quotient module N_{φ} has very rich structure. Indeed, when φ is inner, N_{φ} can be identified with the tensor product of two well-known classical spaces, namely the quotient space H^{2}(Γ) ⊖ φH^{2}(Γ) and the Bergman space L_{a}^{2}(𝔻). Section 5 makes a detailed study of this case.

Acknowledgements. This paper was finished when the second author was visiting the Niigata University. The hospitality and conveniences provided by its Department of Mathematics are greatly appreciated.

2. Preliminaries

For every λ ∈ 𝔻, we define a left evaluation operator L(λ) from H^{2}(Γ^{2}) to H^{2}(Γ_{w}) and a right evaluation operator R(λ) from H^{2}(Γ^{2}) to H^{2}(Γ_{z}) by

L(λ)f(w) = f(λ, w), R(λ)f(z) = f(z, λ), f ∈ H^{2}(Γ^{2}).

Clearly, L(λ) and R(λ) are operator-valued analytic functions over 𝔻. Restrictions of L(λ) and R(λ) to quotient spaces N, M ⊖ zM and M ⊖ wM play key roles in the study here. The following lemma is from [4].

Lemma 2.1. The restriction of R(λ) to M ⊖ wM is equivalent to the characteristic operator function for S_{w}.

The following spectral relations are thus clear. Details can be found in [4] and [18].

(a) λ ∈ σ(S_{w}) if and only if R(λ) : M ⊖ wM → H^{2}(Γ_{z}) is not invertible.

(b) dim ker(S_{w} − λI) = dim ker(R(λ)|_{M⊖wM}).

(c) S_{w} − λI has a closed range if and only if R(λ)(M ⊖ wM) is closed.

(d) S_{w} −λI is Fredholm if and only if R(λ)|_{M⊖wM} is Fredholm, and in this

case

ind(S_{w} − λI) = ind(R(λ)|_{M⊖wM}).

Restrictions T_{z}^{∗}|_{M⊖zM} and T_{w}^{∗}|_{M⊖wM} are also important here, and for simplicity they are denoted by D_{z} and D_{w}, respectively. Clearly,

D_{z}f(z, w) = \frac{f(z, w) − f(0, w)}{z}, D_{w}f(z, w) = \frac{f(z, w) − f(z, 0)}{w},

and it is not hard to check that the ranges of D_{z} and D_{w} are subspaces of N. The following lemma (cf. [22]) gives a description of the defect operators for S_{z}, and it will be used often.

Lemma 2.2. On a quotient module N:

(i) S_{z}^{∗}S_{z} + D_{z}D_{z}^{∗} = I. (ii) S_{z}S_{z}^{∗} + (L(0)|_{N})^{∗}L(0)|_{N} = I.

A parallel version of Lemma 2.2 for S_{w} will also be used.

The operator D_{z} is a useful tool in this study. We first note that

D_{z}^{∗}f = P_{M}zf, f ∈ N.

So if D_{z}^{∗}f = 0, then zf ∈ N. Clearly zf ∈ ker L(0)|_{N}. Conversely, if h is in ker L(0)|_{N}, then we can write h = zh_{0}. One checks easily that h_{0} ∈ ker D_{z}^{∗}. This observation shows that

z ker D_{z}^{∗} = ker L(0)|_{N}.

So on N_{φ}, since L(0)|_{N_{φ}} is injective (cf. [14]), D_{z}^{∗} has trivial kernel, i.e., the range R(D_{z}) is dense in N_{φ}. The following theorem describes R(D_{z}) in detail.

Theorem 2.3. Let N be a quotient module of H^{2}(Γ^{2}) and M = H^{2}(Γ^{2})⊖N. Suppose that L(0)|_{N} is one to one and R(D_{z}) is dense in N. Let f ∈ N. Then f ∈ R(D_{z}) if and only if there exists a positive constant C_{f} depending on f such that |⟨S_{z}^{∗}h, f⟩| ≤ C_{f}∥L(0)h∥ for every h ∈ N.

Proof. Suppose that f ∈ R(D_{z}). Let g ∈ M ⊖ zM with T_{z}^{∗}g = f. We have g = zf + L(0)g. Then for h ∈ N,

|⟨S_{z}^{∗}h, f⟩| = |⟨h, zf⟩|
= |⟨h, g − L(0)g⟩|
= |⟨h, L(0)g⟩|
= |⟨L(0)h, L(0)g⟩|
≤ ∥L(0)g∥∥L(0)h∥.

To prove the converse, suppose that there exists a positive constant C_{f} satisfying

|⟨S_{z}^{∗}h, f⟩| ≤ C_{f}∥L(0)h∥

for every h ∈ N. Since L(0) on N is one to one, we have a map Λ defined by

Λ : L(0)N ∋ u(w) → L(0)^{−1}u → ⟨S_{z}^{∗}L(0)^{−1}u, f⟩ ∈ ℂ.

Note that L(0)^{−1}u ∈ N. Obviously, Λ is linear and

|Λu| = |⟨S_{z}^{∗}L(0)^{−1}u, f⟩| ≤ C_{f}∥L(0)L(0)^{−1}u∥ = C_{f}∥u∥.

Hence by the Hahn–Banach theorem, Λ is extendable to a bounded linear functional on H^{2}(Γ_{w}) and there exists v(w) ∈ H^{2}(Γ_{w}) satisfying ⟨u, v⟩ = Λu for every u ∈ L(0)N. We have

⟨u, v⟩ = ⟨S_{z}^{∗}L(0)^{−1}u, f⟩ = ⟨L(0)^{−1}u, zf⟩.

Since v(w) ∈ H^{2}(Γ_{w}), ⟨u, v⟩ = ⟨L(0)^{−1}u, v⟩. Therefore

⟨L(0)^{−1}u, zf − v⟩ = 0

for every u ∈ L(0)N. Since L(0)^{−1}(L(0)N) = N, we get zf − v ⊥ N. Hence zf − v ∈ M. Since v(w) ∈ H^{2}(Γ_{w}), we have T_{z}^{∗}(zf − v) = f ∈ N. This implies that zf − v ∈ M ⊖ zM. Thus we get f ∈ R(D_{z}). □

In the case of N_{φ}, [14] provides a very useful description of the functions in the space. Let φ(w) ∈ H^{2}(Γ_{w}). For f(w) ∈ H^{2}(Γ_{w}), we formally define a function

(T_{φ}^{∗}f)(w) = ∑_{n=0}^{∞} a_{n}w^{n},

where

a_{n} = ∫_{0}^{2π} φ(e^{iθ})f(e^{iθ})e^{−inθ}dθ/2π = ⟨f(w), φ(w)w^{n}⟩.

Generally, T_{φ}^{∗}f may not be in H^{2}(Γ_{w}). When T_{φ}^{∗}f ∈ H^{2}(Γ_{w}), we can define T_{φ}^{∗2}f = T_{φ}^{∗}(T_{φ}^{∗}f). Inductively if T_{φ}^{∗n}f ∈ H^{2}(Γ_{w}), we can define T_{φ}^{∗(n+1)}f = T_{φ}^{∗}(T_{φ}^{∗n}f). For convenience, we let

A_{φ}f(z, w) = ∑_{n=0}^{∞} z^{n}T_{φ}^{∗n}f(w)

be an operator defined at every f ∈ H^{2}(Γ_{w}) for which A_{φ}f ∈ H^{2}(Γ^{2}). Then it is shown in [14] that L(0) is one-to-one on N_{φ} and

(2.1)  N_{φ} = {A_{φ}f : f(w) ∈ H^{2}(Γ_{w}), ∑_{n=0}^{∞} ∥T_{φ}^{∗n}f∥^{2} < ∞}.

It is easy to see that L(0)A_{φ}f = f. Moreover by [14, Corollary 2.8], L(0)N_{φ} is dense in H^{2}(Γ_{w}).

The following two lemmas are needed for the study of σ(S_{z}).

Lemma 2.4. Let φ(w), g(w) ∈ H^{2}(Γ_{w}) and ψ(w) ∈ H^{∞}(Γ_{w}). Then

T_{φ}^{∗}T_{ψ}^{∗}g = T_{ψφ}^{∗}g.

Moreover if T_{φ}^{∗}g ∈ H^{2}(Γ_{w}), then T_{ψ}^{∗}T_{φ}^{∗}g = T_{ψφ}^{∗}g.

Proof. Let n ≥ 0. Then by the definitions above,

⟨T_{φ}^{∗}T_{ψ}^{∗}g, z^{n}⟩ = ⟨g, φψz^{n}⟩ = ⟨T_{φψ}^{∗}g, z^{n}⟩.

Thus T_{φ}^{∗}T_{ψ}^{∗}g = T_{φψ}^{∗}g. Suppose that T_{φ}^{∗}g ∈ H^{2}(Γ_{w}). We have φg − T_{φ}^{∗}g ∈

zH^{1}. Hence

⟨T_{ψ}^{∗}T_{φ}^{∗}g, z^{n}⟩ = ⟨T_{φ}^{∗}g, ψz^{n}⟩
= ∫_{0}^{2π} φ(e^{iθ})g(e^{iθ})ψ(e^{iθ})e^{−inθ}dθ/2π
= ⟨g, ψφz^{n}⟩.

Thus we get our assertion. □

Let w_{0} ∈ Ω_{φ}. The following lemma follows easily from the calculation

T_{φ}^{∗} \frac{1}{1 − w_{0}w} = \frac{φ(w_{0})}{1 − w_{0}w} .

Lemma 2.5. For w_{0} ∈ Ω_{φ},we have

\frac{1}{(1 − φ(w_{0})z)(1 − w_{0}w)} ∈ N_{φ}.

3. The spectra of S_{z} and S_{w}

The spectra of S_{z} and S_{w} on N_{φ} is evidently dependent on φ. This section aims to figure out how they are exactly related. Lemma 2.1 and the description in (2.1) are helpful to this end.

Proposition 3.1. φ(𝔻) ∩ 𝔻 ⊂ σ(S_{z}) ⊂ φ(𝔻) ∩ \overline{𝔻}.

Proof. Let w_{0} ∈ φ(𝔻) ∩ 𝔻. Then w_{0} = φ(w_{1}) for some w_{1} ∈ 𝔻 and

S_{z}^{∗}(\frac{1}{(1 − φ(w_{1})z)(1 − w_{1}w)}) = ∑_{n=1}^{∞} (φ(w_{1})^{n}(1 − w_{1}w)^{−1})z^{n−1}

= φ(w_{1})(\frac{1}{(1 − φ(w_{1})z)(1 − w_{1}w)}).

By Lemma 2.5, φ(w_{1}) is a point spectrum of S_{z}^{∗}. Thus we get φ(𝔻) ∩ 𝔻 ⊂ σ(S_{z}).

Let λ ∉ φ(𝔻). Then 1/(φ(w) − λ) ∈ H^{∞}(Γ_{w}). Let F ∈ N_{φ}. We have

S_{1/(φ−λ)}^{∗}F = S_{1/(φ−λ)}^{∗} ∑_{n=0}^{∞}(T_{φ}^{∗n}L(0)F)z^{n}

= ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{1/(φ−λ)}^{∗}L(0)F)z^{n} by Lemma 2.4.

Hence

S_{1/(φ−λ)}^{∗}S_{z−λ}^{∗}F = ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{1/(φ−λ)}^{∗}L(0)S_{z−λ}^{∗}F)z^{n}

= ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{1/(φ−λ)}^{∗}T_{φ−λ}^{∗}L(0)F)z^{n}

= ∑_{n=0}^{∞}(T_{φ}^{∗n}L(0)F)z^{n} by Lemma 2.4

= F.

Also we have

S_{z−λ}^{∗}S_{1/(φ−λ)}^{∗}F

= ∑_{n=1}^{∞}(T_{φ}^{∗n}T_{1/(φ−λ)}^{∗}L(0)F)z^{n−1} − λ ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{1/(φ−λ)}^{∗}L(0)F)z^{n}

= ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{φ}^{∗}T_{1/(φ−λ)}^{∗}L(0)F)z^{n} − λ ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{1/(φ−λ)}^{∗}L(0)F)z^{n}

= ∑_{n=0}^{∞}(T_{φ}^{∗n}T_{(φ−λ)}^{∗}T_{1/(φ−λ)}^{∗}L(0)F)z^{n}

= F.

Thus (S_{z} − λ)^{−1} = S_{1/(φ−λ)} and hence λ ∉ σ(S_{z}).

Since ∥S_{z}∥ ≤ 1, we have our assertion. □

For a submodule M in H^{2}(Γ^{2}), the quotient space M ⊖zM is a wandering subspace for the multiplication by z and we have

M = ∑_{n=0}^{∞} ⊕ z^{n}(M ⊖ zM).

For a fixed λ ∈ 𝔻 and every f ∈ M, we write f = ∑_{j=0}^{∞} z^{j}f_{j} for some unique sequence {f_{j}} in M ⊖ zM. So

f = ∑_{j=0}^{∞} λ^{j}f_{j} + ∑_{j=0}^{∞}(z^{j} − λ^{j})f_{j},

which means that f = h_{1} +(z −λ)h_{2} for some h_{1} ∈ M ⊖zM and h_{2} ∈ M. If h_{1}+(z−λ)h_{2} = 0, then h_{1}+zh_{2} = λh_{2}, and hence |λ|^{2}∥h_{2}∥^{2} = ∥h_{1}∥^{2}+∥h_{2}∥^{2}, which is possible only if h_{1} = h_{2} = 0. This observation shows that M can be expressed as the direct sum

(3.1)  M = (M ⊖ zM) ∔ (z − λ)M.

We now look at the spectral properties of S_{w}.

Proposition 3.2. On N_{φ}:

(i) \overline{Ω}_{φ} ⊂ σ(S_{w}). (ii) S_{w} − αI is Fredholm for every α ∈ Ω_{φ} and ind(S_{w} − αI) = −1.

Proof. We use Lemma 2.1 to this end.

(i) It is sufficient to show Ω_{φ} ⊂ σ(S_{w}). If α ∈ Ω_{φ}, then for any function (z−φ)h(z, w) in M_{φ}⊖wM_{φ}, (z−φ(α))h(z, α) vanishes at φ(α), and therefore R(α)(M_{φ} ⊖ wM_{φ}) ⊂ (z − φ(α))H^{2}(Γ_{z}) ≠ H^{2}(Γ_{z}). By Lemma 2.1, α ∈ σ(S_{w}).

(ii) It is equivalent to show that R(α)|_{M_{φ}⊖wM_{φ}} is Fredholm with index −1. We first show that R(α) is injective on M_{φ} ⊖ wM_{φ} for every α ∈ Ω_{φ}. Let (z−φ)h(z, w) be in M_{φ}. Then there is a sequence of polynomials {p_{n}(z, w)}_{n} such that (z −φ)p_{n} converges to (z −φ)h in the norm of H^{2}(Γ^{2}). Since R(α) is a bounded operator, (z − φ(α))p_{n}(z, α) converges to (z − φ(α))h(z, α), which, by the fact |φ(α)| < 1, implies that p_{n}(z, α) converges to h(z, α) in H^{2}(Γ_{z}). Since for every f ∈ H^{2}(Γ_{z}), we have ∥φf∥ = ∥φ∥∥f∥ and hence

(3.2)  ∥(z − φ)f∥ ≤ ∥zf∥ + ∥φf∥ = (1 + ∥φ∥)∥f∥ < ∞,

so (z − φ)p_{n}(z, α) converges to (z − φ)h(z, α) in M_{φ}. It follows that

lim_{n→∞}(z − φ)\frac{p_{n} − p_{n}(·, α)}{w − α} = (z − φ)\frac{h − h(·, α)}{w − α},

which implies that (z − φ)\frac{h−h(·,α)}{w−α} ∈ M_{φ}. If (z − φ)h(z, w) is in M_{φ} ⊖ wM_{φ} such that (z − φ(α))h(z, α) = 0, then h(z, α) = 0, and it follows from the observation above that

(z − φ)h = (w − α)(z − φ)\frac{h}{w − α} ∈ (w − α)M_{φ},

and hence by (3.1) (z − φ)h(z, w) = 0 which implies that R(α) is injective on M_{φ} ⊖ wM_{φ}.

In the proof of (i), we showed that R(α)(M_{φ} ⊖ wM_{φ}) ⊂ (z−φ(α))H^{2}(Γ_{z}). On the other hand, for every g ∈ H^{2}(Γ_{z}), (z − φ)g is in M_{φ} by (3.2), and by (3.1)

(z − φ(α))g ∈ R(α)(M_{φ}) = R(α)(M_{φ} ⊖ wM_{φ}).

This shows that

R(α)(M_{φ} ⊖ wM_{φ}) = (z − φ(α))H^{2}(Γ_{z}),

i.e., R(α)|_{M_{φ}⊖wM_{φ}} has a closed range with codimension 1, and this completes the proof in view of Lemma 2.1. □

Corollary 3.3. If φ is bounded with ∥φ∥_{∞} ≤ 1, then σ(S_{w}) = \overline{𝔻} and σ_{e}(S_{w}) = Γ.

Proof. By Proposition 3.2 and the fact that S_{w} is a contraction, σ(S_{w}) = \overline{𝔻} and σ_{e}(S_{w}) ⊂ Γ. Since ind(S_{w}) = −1, σ_{e}(S_{w}) is a closed curve, and therefore σ_{e}(S_{w}) = Γ. □

We will mention another somewhat deeper consequence of Proposition 3.2 near the end of this section. Here we continue to study the Fredholmness of S_{z}. Unfortunately, the techniques used for Proposition 3.2(ii) can not be applied directly to the case here and a technical difficulty seems hard to overcome. So instead we use (3.1) in this case. We begin with some simple observations.

Lemma 3.4. Let φ(w) = b(w)h(w) be the inner-outer factorization of φ(w). Then ker S_{z}^{∗} = H^{2}(Γ_{w}) ⊖ b(w)H^{2}(Γ_{w}).

Proof. Since the functions in H^{2}(Γ_{w}) ⊖ b(w)H^{2}(Γ_{w}) depend only on w, the inclusion

H^{2}(Γ_{w}) ⊖ b(w)H^{2}(Γ_{w}) ⊂ ker S_{z}^{∗}

is easy to check.

If f is a function in N_{φ} such that S_{z}^{∗}f = 0, then zf is orthogonal to H^{2}(Γ^{2}) which means f is independent of the variable z. Since for every nonnegative integer j

0 = ⟨(z − φ)w^{j}, f⟩ = ⟨−φw^{j}, f⟩,

f is in H^{2}(Γ_{w}) ⊖ b(w)H^{2}(Γ_{w}). □

Theorem 3.5. Let φ(w) = b(w)h(w) be the inner-outer factorization of φ and

α = inf_{w∈𝔻}|h(w)|.

Then S_{z}^{∗} has a closed range if and only if α ≠ 0, and in this case S_{z}^{∗}N_{φ} = N_{φ}.

Proof. Write K_{b} = H^{2}(Γ_{w}) ⊖ b(w)H^{2}(Γ_{w}). By Lemma 3.4, ker S_{z}^{∗} = K_{b}.

Suppose that α > 0. Then h(w)^{−1} ∈ H^{∞}(Γ_{w}) and ∥T_{h^{−1}}^{∗}∥ = ∥h^{−1}∥_{∞} = α^{−1}. Let F ∈ N_{φ} ⊖ K_{b}. We can write (L(0)F)(w) = b(w)f(w). Then by (2.1),

∥F ∥^{2} = ∥ ∑_{n=0}^{∞} z^{n}T_{φ}^{∗n}bf∥^{2}

= ∑_{n=0}^{∞} ∥T_{φ}^{∗n}bf∥^{2}

≥ ∥f∥^{2} + ∥T_{φ}^{∗}bf∥^{2}

= ∥f∥^{2} + ∥T_{h}^{∗}f∥^{2}

= ∥f∥^{2} + α^{2}α^{−2}∥T_{h}^{∗}f∥^{2}

= ∥f∥^{2} + α^{2}∥T_{h^{−1}}^{∗}∥^{2}∥T_{h}^{∗}f∥^{2}

≥ ∥f∥^{2} + α^{2}∥f∥^{2} by Lemma 2.4

= (1 + α^{2})∥L(0)F ∥^{2}.

Since by Lemma 2.2 ∥S_{z}^{∗}F ∥^{2} + ∥L(0)F ∥^{2} = ∥F ∥^{2},

∥S_{z}^{∗}F ∥^{2} = ∥F ∥^{2} − ∥L(0)F ∥^{2} ≥ (1 − \frac{1}{1 + α^{2}} )∥F ∥^{2} = \frac{α^{2}}{1 + α^{2}} ∥F ∥^{2}.

This implies that S_{z}^{∗} is bounded below on N_{φ} ⊖ K_{b}, and hence S_{z}^{∗} has a closed range.

Suppose that α = 0. Let {w_{k}}_{k} be a sequence in 𝔻 satisfying |h(w_{k})| < 1 and h(w_{k}) → 0 as k → ∞. Let

F_{k}(z, w) = \frac{b(w)}{1 − w_{k}w} + ∑_{n=1}^{∞} z^{n} \frac{b(w_{k})^{(n−1)} h(w_{k})^{n}}{1 − w_{k}w} .

Then

∥F_{k}∥^{2} ≥ ∥\frac{1}{1 − w_{k}w} ∥^{2}.

Using the fact that T_{g}^{∗}(1/(1 − w_{k}w)) = g(w_{k})(1/(1 − w_{k}w)) for every g ∈ H^{2}(Γ_{w}), we have

F_{k}(z, w) = ∑_{n=0}^{∞} z^{n}T_{φ}^{∗n} \frac{b(w)}{1 − w_{k}w} ∈ N_{φ} ⊖ K_{b},

and therefore

S_{z}^{∗}F_{k} = ∑_{n=0}^{∞} z^{n} \frac{b(w_{k})^{n} h(w_{k})^{(n+1)}}{1 − w_{k}w} ,

and

∥S_{z}^{∗}F_{k}∥^{2} ≤ ∥\frac{1}{1 − w_{k}w} ∥^{2} \frac{|h(w_{k})|^{2}}{1 − |h(w_{k})|^{2}} .

It follows

∥S_{z}^{∗}F_{k}∥^{2} ≤ \frac{|h(w_{k})|^{2}}{1 − |h(w_{k})|^{2}} ∥F_{k}∥^{2}.

This implies that S_{z}^{∗} is not bounded below on N_{φ} ⊖ K_{b}. Since S_{z}^{∗} is oneto-one on N_{φ} ⊖ K_{b}, S_{z}^{∗}(N_{φ} ⊖ K_{b}) is not a closed subspace. Since S_{z}^{∗}(N_{φ}) = S_{z}^{∗}(N_{φ} ⊖ K_{q}), S_{z}^{∗} does not have a closed range.

Next we shall prove that S_{z}^{∗}N_{φ} = N_{φ} when α > 0. Let g(w) ∈ L(0)N_{φ}. We have

∑_{n=0}^{∞} ∥T_{φ}^{∗n}T_{h^{−1}}^{∗}bg∥^{2} = ∥T_{h^{−1}}^{∗}bg∥^{2} + ∑_{n=1}^{∞} ∥T_{φ}^{∗(n−1)}g∥^{2}
≤ ∥h^{−1}∥_{∞}^{2}∥g∥^{2} + ∥L(0)^{−1}g∥^{2}
< ∞.

Hence T_{h^{−1}}^{∗}bg ∈ L(0)N_{φ}, and

S_{z}^{∗}L(0)^{−1}T_{h^{−1}}^{∗}bg = ∑_{n=1}^{∞} z^{n−1}T_{φ}^{∗n}T_{h^{−1}}^{∗}bg

= ∑_{n=1}^{∞} z^{n−1}T_{φ}^{∗(n−1)}g

= L(0)^{−1}g.

This implies that S_{z}^{∗}N_{φ} = N_{φ}. □

Corollary 3.6. With notations as in Theorem 3.5, the following conditions are equivalent.

(i) α ≠ 0. (ii) S_{z}^{∗} has a closed range. (iii) S_{z}^{∗}N_{φ} = N_{φ}. (iv) T_{φ}^{∗}L(0)N_{φ} = L(0)N_{φ}.

Theorem 3.5 in particular shows that S_{z} is injective when α > 0. This is in fact a general phenomenon on N_{φ}. The following fact (cf. [5, p. 85]) is needed to this end.

Lemma 3.7. Let h(w) be an outer function on Γ_{w}. Then there is a sequence of outer functions {h_{k}}_{k} in H^{∞}(Γ_{w}) such that ∥h_{k}h∥_{∞} ≤ 1 and h_{k}h → 1 a.e. on Γ_{w} as k → ∞.

Theorem 3.8. S_{z} is injective on N_{φ}.

Proof. We show that S_{z}^{∗} has a dense range. Let φ(w) = b(w)h(w) be the inner-outer factorization of φ. By Lemma 3.7, there is a sequence {h_{k}}_{k} in H^{∞}(Γ_{w}) such that

(3.3)  ∥h_{k}h∥_{∞} ≤ 1 and h_{k}h → 1 a.e. on Γ_{w} as k → ∞.

Let g(w) ∈ L(0)N_{φ}. By Lemma 2.4, we have

∑_{n=0}^{∞} ∥T_{φ}^{∗n}T_{h_{k}}^{∗}bg∥^{2} = ∥T_{h_{k}}^{∗}bg∥^{2} + ∑_{n=1}^{∞} ∥T_{h_{k}h}^{∗}T_{φ}^{∗(n−1)}g∥^{2}

≤ ∥h_{k}∥_{∞}^{2}∥g∥^{2} + ∑_{n=1}^{∞} ∥T_{φ}^{∗(n−1)}g∥^{2} by (3.3)

= ∥h_{k}∥_{∞}^{2}∥g∥^{2} + ∥L(0)^{−1}g∥^{2}

< ∞.

Hence T_{h_{k}}^{∗}bg ∈ L(0)N_{φ}, and we have

∥S_{z}^{∗}L(0)^{−1}T_{h_{k}}^{∗}bg − L(0)^{−1}g∥^{2} = ∑_{n=0}^{∞} ∥T_{φ}^{∗(n+1)}T_{h_{k}}^{∗}bg − T_{φ}^{∗n}g∥^{2}

= ∑_{n=0}^{∞} ∥T_{h_{k}h−1}^{∗}T_{φ}^{∗n}g∥^{2}

≤ ∑_{n=0}^{∞} ∥(h_{k}h − 1)T_{φ}^{∗n}g∥^{2}

= ∫_{0}^{2π} |(hh_{k})(e^{iθ}) − 1|^{2} ∑_{n=0}^{∞} |(T_{φ}^{∗n}g)(e^{iθ})|^{2} \frac{dθ}{2π} .

Since g ∈ L(0)N_{φ},

∑_{n=0}^{∞} |T_{φ}^{∗n}g|^{2} ∈ L^{1}(Γ_{w}).

Hence by (3.3) and the Lebesgue dominated convergence theorem,

∥S_{z}^{∗}L(0)^{−1}T_{h_{k}}^{∗}bg − L(0)^{−1}g∥^{2} → 0 as k → ∞.

This implies that S_{z}^{∗} has a dense range. □

Corollary 3.9. Let φ(w) = b(w)h(w) be the inner-outer factorization of φ(w). Then the following are equivalent.

(i) S_{z} is Fredholm. (ii) b(w) is a finite Blaschke product and h^{−1}(w) ∈ H^{∞}(Γ_{w}). In this case, − ind(S_{z}) is the number of zeros of b(w) in 𝔻 counting multiplicites.

Proof. We let α = inf_{w∈𝔻} |h(w)|. S_{z} is Fredholm if and only if S_{z}^{∗} is Fredholm, and by Lemma 3.4 and Theorem 3.5 this is equivalent to b being a finite Blaschke product and α > 0. Clearly, α > 0 if and only if h^{−1}(w) ∈ H^{∞}(Γ_{w}). □

A quotient module N is said to be essentially reductive if both S_{z} and S_{w} are essentially normal, i.e., [S_{z}^{∗}, S_{z}] and [S_{w}^{∗}, S_{w}] are both compact. Essential reductivity is an important concept and has been studied recently in various contexts. In the context here, it will be interesting to see what type of φ makes N_{φ} essentially reductive. Proposition 3.2 has a couple of consequences to this end. A general study will be made in a different paper.

Corollary 3.10. For every φ ∈ H^{2}(Γ_{w}), [S_{z}^{∗}, S_{w}] is Hilbert–Schmidt on N_{φ}.

Proof. We let R_{z} and R_{w} denote the multiplications by z and w on the submodule M_{φ}, respectively. It then follows from Proposition 3.2 and Theorem 2.3 in [21] that [R_{z}^{∗}, R_{z}][R_{w}^{∗}, R_{w}] is Hilbert–Schmidt, and the corollary thus follows from Theorem 2.6 in [21]. □

In the case φ is in the disk algebra A(𝔻), there is a sequence of polynomials {p_{n}}_{n} satisfying p_{n} → φ in A(𝔻), and hence [S_{z}^{∗}, p_{n}(S_{w})] → [S_{z}^{∗}, φ(S_{w})] in operator norm. Since S_{z} = φ(S_{w}) on N_{φ}, we easily obtain the following corollary.

Corollary 3.11. If φ ∈ A(𝔻), then S_{z} is essentially normal.

Question 1. For what φ ∈ H^{2}(Γ_{w}) is S_{w} essentially normal on N_{φ}?

In the case φ is inner, this question can be settled by direct calculations. We will do it in Section 5.

4. Compactness of L(0)|_{N_{φ}} and D_{z}

In view of Lemma 2.2, the compactness of L(0)|_{N} or D_{z} will give us much information about the operator S_{z}. So to determine whether L(0)|_{N} or D_{z} is compact for a certain quotient module N is of great interest. In the case of N_{φ}, the compactness is undoubtly dependent on the properties of φ. This section aims to unveil the connection.

We first look at the compactness of L(0)|_{Nφ}. For each fixed ζ ∈ 𝔻, we denote by Z_{φ}(ζ) the number of zeros of ζ −φ(w) in 𝔻 counting multiplicities. This integer-valued function has an important role to play in this study. As a matter of fact, in [22, Theorem 5.2.2], the second author showed that if L(0) on N_{φ} is compact, then Z_{φ}(ζ) is a finite constant on 𝔻. The following describes the functions φ for which this is the case.

Lemma 4.1. Let φ(w) = b(w)h(w) be the inner-outer factorization of φ. Then Z_{φ}(ζ) is a finite constant on 𝔻 if and only if b is a finite Blaschke product and |h(w)| ≥ 1 for every w ∈ 𝔻.

Proof. It is easy to see that that b is a finite Blaschke product and |h(w)| ≥ 1 for every w ∈ 𝔻 if and only if

lim inf_{|w|→1} |φ(w)| ≥ 1.

Suppose that c = Z_{φ}(ζ) for every ζ ∈ 𝔻. To prove the necessity by contradiction, we assume that there exists a sequence {w_{n}}_{n} in 𝔻 such that sup_{n} |φ(w_{n})| < 1 and |w_{n}| → 1. We may assume that φ(w_{n}) → ζ_{0} ∈ 𝔻. Then there exists r_{0}, 0 < r_{0} < 1, such that the number of zeros of ζ_{0} − φ(w) in r_{0}𝔻 is equal to c. By the Hurwitz theorem, for a large positive integer n_{0}, the number of zeros of φ(w_{n_{0}}) − φ(w) in r_{0}𝔻 is equal to c. Further, we may assume that w_{n_{0}} ∉ r_{0}𝔻. Hence the number of zeros of φ(w_{n_{0}}) − φ(w) in 𝔻 is greater than c which contradicts the fact that Z_{φ}(ζ) is a constant.

The sufficiency is an easy consequence of Rouché’s theorem in complex analysis. In fact, if b(w) is a finite Blaschke product and h(w) is an outer function with |h(w)| ≥ 1 on 𝔻, then by Rouché’s theorem, for each ζ ∈ 𝔻 the number of zeros of ζ − φ(w) in 𝔻 coincides with the number of zeros of b(w) in 𝔻. So Z_{φ}(ζ) is a finite constant. □

Theorem 4.2. Let φ(w) = b(w)h(w) be the inner-outer factorization of φ. Then the following conditions are equivalent.

(i) L(0) on N_{φ} is compact. (ii) b is a finite Blaschke product and |h(w)| ≥ 1 for every w ∈ 𝔻.

Proof. (i) ⇒ (ii) If L(0) on N_{φ} is compact, then by Theorem 5.2.2 in [22] Z_{φ}(ζ) is a finite constant, and (ii) thus follows from Lemma 4.1.

(ii) ⇒ (i) Since b is a finite Blaschke product, for any positive integer m, we have dim (H^{2}(Γ_{w}) ⊖ b^{m}(w)H^{2}(Γ_{w})) < ∞ and H^{2}(Γ_{w}) ⊖ b^{m}(w)H^{2}(Γ_{w})

444 K. Izuchi and R. Yang is contained in the disk algebra A(𝔻). One easily sees that

T_{φ}^{∗j}(H^{2}(Γ_{w}) ⊖ b^{m}(w)H^{2}(Γ_{w})) = {0}, j > m,

so that

H^{2}(Γ_{w}) ⊖ b^{m}(w)H^{2}(Γ_{w}) ⊂ L(0)N_{φ}.

Then

L(0)N_{φ} = (H^{2}(Γ_{w}) ⊖ b^{m}H^{2}(Γ_{w})) ⊕ (b^{m}H^{2}(Γ_{w}) ∩ L(0)N_{φ})

and hence

N_{φ} = L(0)^{−1}(H^{2}(Γ_{w}) ⊖ b^{m}H^{2}(Γ_{w})) ∔ L(0)^{−1}(b^{m}H^{2}(Γ_{w}) ∩ L(0)N_{φ}), which is in fact a direct sum because L(0)|_{N_{φ}} is injective. For simplicity we write this decomposition as

N_{φ} = N_{1,m} ∔ N_{2,m}.

Since dim(N_{1,m}) < ∞, to prove that L(0) on N_{φ} is compact it is sufficient to prove that lim_{m→∞} ∥L(0)|_{N_{2,m}}∥ = 0, i.e.,

sup_{b^{m}g∈L(0)N_{φ}} \frac{∥b^{m}g∥^{2}}{∥L(0)^{−1}b^{m}g∥^{2}} → 0 as m → ∞.

Let b^{m}g ∈ L(0)N_{φ} and 0 ≤ n ≤ m. By Lemma 2.4, T_{h}^{∗}b^{m−1}g = T_{φ}^{∗}b^{m}g ∈ H^{2}(Γ_{w}), so that

T_{h}^{∗2}b^{m−2}g = T_{h}^{∗}T_{h}^{∗}T_{b}^{∗}b^{m−1}g = T_{h}^{∗}T_{b}^{∗}T_{h}^{∗}b^{m−1}g = T_{φ}^{∗2}b^{m}g ∈ H^{2}(Γ_{w}). Repeating this, we have

(4.1)  T_{h}^{∗n}b^{m−n}g = T_{φ}^{∗n}b^{m}g ∈ H^{2}(Γ_{w}).

Using the fact that L(0)A_{φ}f = f, i.e.,

L(0)^{−1}f = ∑_{j=0}^{∞} z^{j}T_{φ}^{∗j}f,

and that ∥h^{−1}∥_{∞} ≤ 1, we calculate that

sup_{b^{m}g∈L(0)N_{φ}} \frac{∥b^{m}g∥^{2}}{∥L(0)^{−1}b^{m}g∥^{2}} = sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{j=0}^{∞} ∥T_{φ}^{∗j}b^{m}g∥^{2}}

≤ sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{j=0}^{m} ∥T_{φ}^{∗j}b^{m}g∥^{2}}

= sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{j=0}^{m} ∥T_{h}^{∗j}b^{m−j}g∥^{2}} by (4.1)

≤ sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{j=0}^{m} ∥T_{h^{−1}}^{∗j}∥^{2}∥T_{h}^{∗j}b^{m−j}g∥^{2}}

≤ sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{j=0}^{m} ∥b^{m−j}g∥^{2}} by Lemma 2.4

= \frac{1}{m + 1}.

So it follows that lim_{m→∞} ∥L(0)|_{N_{2,m}}∥ = 0 and this completes the proof. □

Corollary 4.3. If L(0) and R(0) are both compact on N_{φ} then φ is a finite Blaschke product.

Proof. If R(0) is compact on N_{φ}, then by the parallel statement of Theorem 5.2.2 in [22] for R(0), the number of zeros of z − φ(λ) in 𝔻 is a constant with respect to λ ∈ 𝔻. Since N_{φ} is nontrivial, this constant is equal to 1. So ∥φ∥_{∞} ≤ 1, and it follows that ∥h∥_{∞} ≤ 1. If L(0) is also compact on N_{φ}, then by Theorem 4.2 h is a constant of modulous 1, hence φ is a finite Blaschke product. □

In fact the converse of Corollary 4.3 is also true and we will see it in Section 5.

Next we study the compactness of D_{z}. In fact, the compactness of D_{z} and that of L(0)|_{N_{φ}} are closely related.

Theorem 4.4. If φ is bounded, then L(0)|_{N_{φ}} is compact if and only if D_{z} is compact.

Proof. The fact that the compactness of L(0)|_{N_{φ}} implies the compactness of D_{z} follows from Theorem 3.8 and [22, Theorem 5.3.1].

To show that the compactness of D_{z} implies that of L(0)|_{N_{φ}}, we first check that S_{z} is Fredholm in this case. If D_{z} is compact, then by Lemma 2.2 S_{z}^{∗}S_{z} is Fredholm, and hence S_{z}^{∗} has closed range. Moreover, it follows from Theorem 3.8 that S_{z}^{∗} is in fact onto. So it remains to show that S_{z}^{∗} has a finite-dimensional kernel. If we let φ = bh be the inner-outer factorization of φ, then by Lemma 3.4 we need to show that H^{2}(Γ_{w}) ⊖ bH^{2}(Γ_{w}) is a finite-dimensional subspace in N_{φ}, or equivalently, b is a Blaschke product.

For every f ∈ H^{2}(Γ_{w}) ⊖ bH^{2}(Γ_{w}) and integers i, j ≥ 0, one checks that

⟨D_{z}^{∗}f, (z − φ)z^{i}w^{j}⟩ = ⟨zf, (z − φ)z^{i}w^{j}⟩ = ⟨f, z^{i}w^{j}⟩.

So D_{z}^{∗}f is orthogonal to (z − φ)z^{i}w^{j} when i ≥ 1. Therefore,

∥D_{z}^{∗}f∥ = ∥P_{M_{φ}}zf∥

≥ sup_{∥(z−φ)p∥≤1} |⟨zf, (z − φ)p⟩|, p is polynomial in H^{2}(Γ_{w})

= sup_{∥(z−φ)p∥≤1} |⟨f, p⟩|.

Since

∥(z − φ)p∥^{2} = ∥p∥^{2} + ∥φp∥^{2} ≤ ∥p∥^{2}(1 + ∥φ∥_{∞}^{2}),

we have

∥D_{z}^{∗}f∥ ≥ sup_{∥p∥≤(1+∥φ∥_{∞}^{2})^{−1/2}} |⟨f, p⟩| = (1 + ∥φ∥_{∞}^{2})^{−1/2}∥f∥,

which means D_{z}^{∗} is bounded below by a positive constant on H^{2}(Γ_{w}) ⊖ bH^{2}(Γ_{w}). Since D_{z} is compact, H^{2}(Γ_{w}) ⊖ bH^{2}(Γ_{w}) is finite-dimensional, and we conclude that S_{z} is Fredholm.

Now we show that L(0)|_{N_{φ}} is compact. For this, we recall the equality (cf. Proposition 5.1.1 in [22])

S_{z}D_{z} + (L(0)|_{N_{φ}})^{∗}(L(0)|_{M_{φ}⊖zM_{φ}}) = 0.

Since D_{z} is compact, (L(0)|_{N_{φ}})^{∗}(L(0)|_{M_{φ}⊖zM_{φ}}) is compact. Since we have shown that S_{z} is Fredholm in this case, L(0)|_{M_{φ}⊖zM_{φ}} is Fredholm by Lemma 2.1, and therefore L(0)|_{N_{φ}} is compact. □

The following example gives a simple illustration for the compactness of L(0)|_{N_{φ}}.

Example 1. We consider a function φ(w) = aw, where a ∈ ℂ and a ≠ 0. Let

R_{j} = \sqrt{1 + |a|^{2} + ⋯ + |a|^{2j}}

and

e_{j} = \frac{w^{j} + (az)w^{j−1} + ⋯ + (az)^{j}}{R_{j}} .

Then it is not difficult to check that {e_{j}}_{j} is an orthonormal basis of N_{φ}, and one verifies that

∥L(0)e_{j}∥^{2} = ∥\frac{w^{j}}{R_{j}} ∥^{2} = R_{j}^{−2}.

So if |a| < 1, then ∥L(0)e_{j}∥^{2} ≥ 1 − |a|^{2} and hence L(0) on N_{φ} is not compact. If |a| ≥ 1, then lim_{j→∞} ∥L(0)e_{j}∥ = 0 which shows that L(0) on N_{φ} is compact.

It is clear by Corollary 3.11 that S_{z} is essentially normal in this case. It is easy to give a direct calculation of [S_{z}^{∗}, S_{z}]. In fact,

S_{z}e_{j} = \frac{aR_{j}}{R_{j+1}} e_{j+1}, S_{z}^{∗}e_{j} = \frac{aR_{j−1}}{R_{j}} e_{j−1},

so

(S_{z}^{∗}S_{z} − S_{z}S_{z}^{∗})e_{j} = |a|^{2}(\frac{R_{j}^{2}}{R_{j+1}^{2}} − \frac{R_{j−1}^{2}}{R_{j}^{2}} )e_{j}

= (\frac{|a|^{2} + ⋯ + |a|^{2(j+1)}}{1 + |a|^{2} + ⋯ + |a|^{2(j+1)}} − \frac{|a|^{2} + ⋯ + |a|^{2j}}{1 + |a|^{2} + ⋯ + |a|^{2j}} )e_{j}

:= c_{j}e_{j}.

It is clear that c_{j} → 0 as j → ∞. One also observes that S_{z} on N_{aw} is hyponormal.

By [14], we know that ∥S_{z}∥ = ∥φ∥_{∞} if ∥φ∥_{∞} ≤ 1, and ∥S_{z}∥ = 1 for other cases. In the last part of this section, we calculate the norm and the essential norm of L(0)|_{N_{φ}} and S_{z}. First we recall that the essential norm ∥A∥_{e} is the norm of A in the Calkin algebra.

Since ∥S_{z}^{∗}F ∥^{2} + ∥L(0)F ∥^{2} = ∥F ∥^{2} for every F ∈ N_{φ}, we have

∥S_{z}^{∗}∥^{2} = sup_{F ∈N_{φ},∥F ∥=1} ∥S_{z}^{∗}F ∥^{2} = 1 − inf_{F ∈N_{φ},∥F ∥=1} ∥L(0)F ∥^{2}

and

(4.2) inf_{F ∈N_{φ},∥F ∥=1} ∥S_{z}^{∗}F ∥^{2} = 1 − sup_{F ∈N_{φ},∥F ∥=1} ∥L(0)F ∥^{2} = 1 − ∥L(0)|_{N_{φ}}∥^{2}.

Hence

inf_{F ∈N_{φ},∥F ∥=1} ∥L(0)F ∥ = \{ \begin{matrix}\sqrt{1 − ∥φ∥_{∞}^{2}}, & if ∥φ∥_{∞} ≤ 1 \\ 0, & otherwise. \end{matrix}

Proposition 4.5. Let α = inf_{w∈𝔻} |φ(w)|. Then α < 1 and

∥L(0)|_{N_{φ}}∥ = \sqrt{1 − α^{2}}.

Proof. By [14, Corollary 2.7], φ(𝔻) ∩ 𝔻 ≠ ∅. Hence α < 1. Let w_{0} ∈ Ω_{φ} and

F = \frac{2}{(1 − φ(w_{0})z)(1 − w_{0}w)}.

Then by Lemma 2.5, F ∈ N_{φ} and

\frac{∥L(0)F ∥^{2}}{∥F ∥^{2}} = 1 − |φ(w_{0})|^{2}.

This implies 1 − |φ(w_{0})|^{2} ≤ ∥L(0)|_{N_{φ}}∥^{2}. Thus we get

(4.3)  \sqrt{1 − α^{2}} ≤ ∥L(0)∥ ≤ 1.

If α = 0, then ∥L(0)|_{N_{φ}}∥ = 1.

Suppose that α > 0. Then (1/φ)(w) ∈ H^{∞}(Γ_{w}), and by Lemma 2.4 we have T_{1/φ^{n}}^{∗}T_{φ}^{∗n} = I on L(0)N_{φ} for every n ≥ 0. Let h ∈ L(0)N_{φ}. We have

∥h∥ = ∥T_{1/φ^{n}}^{∗}T_{φ}^{∗n}h∥
≤ ∥T_{1/φ^{n}}^{∗}∥∥T_{φ}^{∗n}h∥
= ∥1/φ∥_{∞}^{n}∥T_{φ}^{∗n}h∥
= ∥T_{φ}^{∗n}h∥/α^{n}.

Then α^{n}∥h∥ ≤ ∥T_{φ}^{∗n}h∥ for every h ∈ L(0)N_{φ} and n. Hence

∥h∥^{2} \frac{1}{1 − α^{2}} ≤ ∑_{n=0}^{∞} ∥T_{φ}^{∗n}h∥^{2} = ∥L(0)^{−1}h∥^{2}

for every h ∈ L(0)N_{φ}, and ∥L(0)F ∥^{2} ≤ (1 − α^{2})∥F ∥ for every F ∈ N_{φ}. Therefore ∥L(0)|_{N_{φ}}∥ ≤ \sqrt{1 − α^{2}}. By (4.3), ∥L(0)|_{N_{φ}}∥ = \sqrt{1 − α^{2}}. □

A combination of (4.2), Propositions 3.1 and 4.5 leads to the following.

Corollary 4.6. Let α = inf_{w∈𝔻} |φ(w)|. Then S_{z}^{∗} is invertible if and only if α > 0. In this case,

∥S_{z}^{∗−1}∥^{−1} = inf_{F ∈N_{φ},∥F ∥=1} ∥S_{z}^{∗}F ∥ = α.

For ζ ∈ Ω_{φ}, let

k_{ζ}(z, w) = \frac{\sqrt{1 − |φ(ζ)|^{2}}}{1 − φ(ζ)z} \frac{\sqrt{1 − |ζ|^{2}}}{1 − ζw} .

By Lemma 2.5, k_{ζ} ∈ N_{φ} and ∥k_{ζ}∥ = 1.

Theorem 4.7. Let φ(w) ∈ H^{2}(Γ_{w}) and φ(w) = b(w)h(w) be the outerinner factorization of φ. Suppose that L(0) on N_{φ} is not compact. Let γ = lim inf_{|w|→1} |φ(w)|. Then γ < 1 and ∥L(0)|_{N_{φ}}∥_{e} = \sqrt{1 − γ^{2}}. Moreover ∥L(0)|_{N_{φ}}∥_{e} ≠ ∥L(0)|_{N_{φ}}∥ if and only if b(w) is a nonconstant finite Blaschke product and 1/h(w) ∈ H^{∞}(Γ_{w}).

Proof. By Theorem 4.2, γ < 1. Take a sequence {w_{j}}_{j} in Ω_{φ} such that |φ(w_{j})| → γ and |w_{j}| → 1 as j → ∞. We have

∥L(0)k_{w_{j}}∥ = \sqrt{1 − |w_{j}|^{2}}\sqrt{1 − |φ(w_{j})|^{2}}∥\frac{1}{1 − w_{0}w} ∥
= \sqrt{1 − |φ(w_{j})|^{2}}
→ \sqrt{1 − γ^{2}}.

Let K be a compact operator from N_{φ} to H^{2}(Γ_{w}). Since k_{w_{j}} → 0 weakly in N_{φ}, ∥(L(0) + K)k_{w_{j}}∥ → \sqrt{1 − γ^{2}}. Hence ∥L(0)|_{N_{φ}}∥_{e} ≥ \sqrt{1 − γ^{2}}.

Suppose that γ = 0. Then 1 ≤ ∥L(0)|_{N_{φ}}∥_{e} ≤ ∥L(0)|_{N_{φ}}∥ ≤ 1. In this case, either b is not a finite Blaschke product or 1/h ∉ H^{∞}(Γ_{w}).

Suppose that 0 < γ < 1. Then b is a finite Blaschke product. By Proposition 4.5, ∥L(0)|_{N_{φ}}∥ = \sqrt{1 − α^{2}}, where α = inf_{w∈𝔻} |φ(w)|. We note that α ≤ γ. If α = γ, then we have ∥L(0)|_{N_{φ}}∥ = ∥L(0)|_{N_{φ}}∥_{e} = \sqrt{1 − γ^{2}}. In this case, b is a constant function and 1/h ∈ H^{∞}(Γ_{w}).

If α < γ, then b is a nonconstant finite Blaschke product and 1/h ∈ H^{∞}(Γ_{w}). This implies that α = 0 and ∥L(0)|_{N_{φ}}∥ = 1. In this case we shall prove that ∥L(0)|_{N_{φ}}∥_{e} = \sqrt{1 − γ^{2}}. We note that ∥1/h∥_{∞} = 1/γ. The idea of the proof is the same as that of Theorem 4.2. We have

sup_{b^{m}g∈L(0)N_{φ}} \frac{∥b^{m}g∥^{2}}{∥L^{−1}(0)b^{m}g∥^{2}} ≤ sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{n=0}^{m} ∥T_{h}^{∗n}b^{m−n}g∥^{2}}

= sup_{b^{m}g∈L(0)N_{φ}} \frac{∥g∥^{2}}{∑_{n=0}^{m} γ^{2n}∥T_{1/h}^{∗n}∥^{2}∥T_{h}^{∗n}b^{m−n}g∥^{2}}

≤ \frac{1}{∑_{n=0}^{m} γ^{2n}} .

Hence ∥L(0)|_{N_{φ}}∥_{e} ≤ \sqrt{1 − γ^{2}}, so that we obtain

∥L(0)|_{N_{φ}}∥_{e} = \sqrt{1 − γ^{2}} < \sqrt{1 − α^{2}} = ∥L(0)|_{N_{φ}}∥.  □

Theorem 4.8. ∥S_{z}∥_{e} = ∥S_{z}∥ for every N_{φ}.

Proof. First, suppose that 0 < ∥φ∥_{∞} ≤ 1. Let K be a compact operator on N_{φ}. Let {w_{j}}_{j} be a sequence in Ω_{φ} such that |φ(w_{j})| → ∥φ∥_{∞} as j → ∞. Then Kk_{w_{j}} → 0 as j → ∞. One easily sees that ∥S_{z}^{∗}k_{w_{j}}∥ = |φ(w_{j})|, so that ∥S_{z}^{∗}k_{w_{j}}∥ → ∥φ∥_{∞} as j → ∞. Hence ∥S_{z}^{∗} + K∥ ≥ ∥φ∥_{∞}. By [14, Proposition 3.5], ∥S_{z}^{∗}∥ = ∥φ∥_{∞}, so that

∥S_{z}∥_{e} = ∥S_{z}^{∗}∥_{e} ≥ ∥φ∥_{∞} = ∥S_{z}^{∗}∥ = ∥S_{z}∥.

Thus we get ∥S_{z}∥_{e} = ∥S_{z}∥.

Next, suppose that 1 < ∥φ∥_{∞} ≤ ∞. By [14, Proposition 3.5], ∥S_{z}∥ = 1. Suppose that lim inf_{|w|→1} |φ(w)| ≥ 1. By Theorem 4.2, L(0) is compact on N_{φ}. Since S_{z}S_{z}^{∗} = I − (L(0)|_{N_{φ}})^{∗}L(0)|_{N_{φ}}, ∥S_{z}S_{z}^{∗}∥_{e} = 1, so that ∥S_{z}∥_{e} = 1.

Suppose that α := lim inf_{|w|→1} |φ(w)| < 1. Take a sequence {w_{j}}_{j} in Ω_{φ} such that lim inf_{j→∞} |φ(w_{j})| = α and |w_{j}| → 1 as j → ∞. Let α_{j} = max_{w∈Γ} |φ(w_{j}w)|. Since ∥φ∥_{∞} > 1, we may assume that α_{j} > 1 for every j. Since |φ(w_{j})| < 1, φ(w_{j}Γ) is a closed curve in ℂ which interesects with both 𝔻 and ℂ \ \overline{𝔻}. Hence there is ζ_{j} ∈ Γ satisfying 1 − 1/j < |φ(w_{j}ζ_{j})| < 1. Note that w_{j}ζ_{j} ∈ Ω_{φ}. Let K be a compact operator on N_{φ}. Then ∥(S_{z}^{∗} + K)k_{w_{j}ζ_{j}}∥ = |φ(w_{j}ζ_{j})| → 1 as j → ∞, so ∥S_{z}^{∗} + K∥ ≥ 1. Hence

∥S_{z}∥_{e} = ∥S_{z}^{∗}∥_{e} ≥ 1 ≥ ∥S_{z}∥ ≥ ∥S_{z}∥_{e}.

Thus we get the assertion. □

5. The case when φ is inner

This section gives a detailed study for the case when φ is inner. On the one hand, the fact that φ is inner makes this case very computable, and, as a consequence, many of the earlier results have a clean illustration in this case. On the other hand, the case has a close connection with the two classical spaces, namely the quotient space H^{2}(Γ) ⊖ φH^{2}(Γ) and the Bergman space L_{a}^{2}(𝔻). This fact suggests that the space N_{φ} indeed has very rich structure.

Some preparations are needed to start the discussion. With every inner function θ(w) in the Hardy space H^{2}(Γ_{w}) over the unit circle Γ_{w}, there is an associated contraction S(θ) on H^{2}(Γ_{w}) ⊖ θH^{2}(Γ_{w}) defined by

S(θ)f = P_{θ}wf, f(w) ∈ H^{2}(Γ_{w}) ⊖ θH^{2}(Γ_{w}),

where P_{θ} is the projection from H^{2}(Γ_{w}) onto H^{2}(Γ_{w}) ⊖ θH^{2}(Γ_{w}). The operator S(θ) is the classical Jordan block, and its properties have been very well studied (cf. [1, 18]). We will state some of the related facts later in the section. Here, we display an orthonormal basis for N_{φ}.

Lemma 5.1. Let φ(w) be a one variable nonconstant inner function. Let {λ_{k}(w)}_{k=0}^{m} be an orthonormal basis of H^{2}(Γ_{w}) ⊖ φ(w)H^{2}(Γ_{w}), and

e_{j} = \frac{w^{j} + w^{j−1}z + ⋯ + z^{j}}{\sqrt{j + 1}}

for each integer j ≥ 0. Then

{λ_{k}(w)e_{j}(z, φ(w)) : k = 0, 1, 2, …, m, j = 1, 2, … }

is an othonormal basis for N_{φ}.

Proof. First of all, we have the facts that

N_{φ} = {A_{φ}f : f ∈ H^{2}(Γ_{w}), ∑_{n=0}^{∞} ∥T_{φ^{n}}^{∗}f∥^{2} < ∞},

and

H^{2}(Γ_{w}) = ∑_{j=0}^{∞} ⊕ φ^{j}(w)(H^{2}(Γ_{w}) ⊖ φ(w)H^{2}(Γ_{w})).

Write

E_{k,j} = λ_{k}(w)e_{j}(z, φ(w)).

Then if (k, j) ≠ (s, t) and j ≤ t,

⟨E_{k,j}, E_{s,t}⟩ = \frac{1}{\sqrt{j + 1}\sqrt{t + 1}} ∑_{l=0}^{j} ∑_{i=0}^{t} ⟨λ_{k}(w)φ^{j−l}(w)z^{l}, λ_{s}(w)φ^{t−i}(w)z^{i}⟩

= \frac{(j + 1)⟨λ_{k}(w), φ^{t−j}(w)λ_{s}(w)⟩}{\sqrt{j + 1}\sqrt{t + 1}}

= 0, and ∥E_{k,j}∥ = 1 for every k, j. Let f(w) ∈ H^{2}(Γ_{w}) and write

f(w) = ∑_{j=0}^{∞} ⊕( ∑_{k=0}^{m} a_{k,j}λ_{k}(w))φ^{j}(w), ∑_{j=0}^{∞} ∑_{k=0}^{m} |a_{k,j}|^{2} < ∞.

Then

∑_{n=0}^{∞} ∥T_{φ^{n}}^{∗}f(w)∥^{2} = ∑_{n=0}^{∞} ∑_{j=n}^{∞} ∑_{k=0}^{m} |a_{k,j}|^{2} = ∑_{j=0}^{∞}(j + 1) ∑_{k=0}^{m} |a_{k,j}|^{2}.

Hence

∑_{n=0}^{∞} z^{n}T_{φ^{n}}^{∗}f(w) ∈ N_{φ} ⟺ ∑_{j=0}^{∞}(j + 1) ∑_{k=0}^{m} |a_{k,j}|^{2} < ∞.

In this case, we have

∑_{n=0}^{∞} z^{n}T_{φ^{n}}^{∗}f(w) = ∑_{j=0}^{∞} ( ∑_{k=0}^{m} a_{k,j}λ_{k}(w))(φ^{j}(w) + φ^{j−1}(w)z + ⋯ + z^{j})

= ∑_{j=0}^{∞} ∑_{k=0}^{m} \sqrt{j + 1}a_{k,j}E_{k,j}.

This shows that {E_{k,j}}_{k,j} is an othonormal basis of N_{φ} = H^{2}(Γ^{2}) ⊖ M_{φ}. □

The operators L(0)|_{N_{φ}}, R(0)|_{N_{φ}} and D_{z} are easy to calculate in this case. In fact, one checks that

L(0)E_{k,j} = \frac{λ_{k}(w)φ^{j}(w)}{\sqrt{j + 1}},

and

R(0)E_{k,j} = \frac{λ_{k}(0)(φ(0)^{j} + φ(0)^{j−1}z + ⋯ + z^{j})}{\sqrt{j + 1}}.

So L(0)|_{N_{φ}} and R(0)|_{N_{φ}} are both compact if m < ∞, that is, φ(w) is a finite Blaschke product. We summarize this observation and Corollary 4.3 in the following corollary.

Corollary 5.2. For φ ∈ H^{2}(Γ_{w}), L(0) and R(0) are both compact on N_{φ} if and only if φ is a finite Blaschke product.

The operator D_{z} is also easy to calculate in this case. One first verifies that

X_{k,j} := \frac{λ_{k}(w)}{\sqrt{j + 2}}(ze_{j}(z, φ(w)) − \sqrt{j + 1}φ^{j+1}(w)), 0 ≤ k ≤ m, 0 ≤ j < ∞,

is an othonormal basis for M_{φ} ⊖ zM_{φ}. Then

(5.1)  D_{z}X_{k,j} = \frac{λ_{k}(w)e_{j}(z, φ(w))}{\sqrt{j + 2}} = \frac{1}{\sqrt{j + 2}}E_{k,j}

which is also compact if φ(w) is a finite Blaschke product.

Two other observations are also worth mentioning. First one calculates that

⟨zE_{k,j}, E_{s,t}⟩ = \frac{1}{\sqrt{j + 1}\sqrt{t + 1}} ∑_{l=0}^{j} ∑_{i=0}^{t} ⟨zλ_{k}(w)φ^{j−l}(w)z^{l}, λ_{s}(w)φ^{t−i}(w)z^{i}⟩

= \frac{1}{\sqrt{j + 1}\sqrt{t + 1}} ∑_{l=0}^{j} ∑_{i=0}^{t} ⟨λ_{k}(w), λ_{s}(w)φ^{t+l−i−j}(w)z^{i−l−1}⟩.

Hence

⟨zE_{k,j}, E_{s,t}⟩ ≠ 0 ⟺ t = j + 1 and k = s,

and

S_{z}E_{k,j} = ⟨S_{z}E_{k,j}, E_{k,j+1}⟩E_{k,j+1}

= \frac{1}{\sqrt{j + 1}\sqrt{j + 2}} ∑_{l=0}^{j}⟨λ_{k}(w), λ_{k}(w)⟩E_{k,j+1}

= \begin{matrix}√ & \overline{j + 1} \\ √ & \overline{j + 2} \end{matrix}E_{k,j+1}.

This calculation reminds us of the Bergman shift B on the Bergman space L_{a}^{2}(𝔻) with the orthonormal basis \{\sqrt{j + 1}ζ^{j}\}_{j}. In fact, if we define the operator

U : N_{φ} ⟶ (H^{2}(Γ) ⊖ φH^{2}(Γ)) ⊗ L_{a}^{2}(𝔻)

by

(5.2)  U(E_{k,j}) = λ_{k}(w)\sqrt{j + 1}ζ^{j},

then U is clearly a unitary operator, and one checks that

(5.3)  US_{z} = (I ⊗ B)U.

So from this view point N_{φ} can be identified as (H^{2}(Γ) ⊖ φH^{2}(Γ)) ⊗ L_{a}^{2}(𝔻). As both H^{2}(Γ) ⊖ φH^{2}(Γ) and L_{a}^{2}(𝔻) are classical subjects, this observation indicates that the space N_{φ} indeed has very rich structure.

The other observation is about the range R(D_{z}). Let F ∈ N_{φ}. Then by Theorem 2.3,

F ∈ D_{z}(M_{φ} ⊖ zM_{φ}) ⟺ sup_{G∈N_{φ},∥G∥=1} \frac{|⟨S_{z}^{∗}G, F ⟩|}{∥L(0)G∥} < ∞.

Write

F = ∑_{k=0}^{m} ∑_{j=0}^{∞} a_{k,j}E_{k,j}, ∑_{k=0}^{m} ∑_{j=0}^{∞} |a_{k,j}|^{2} < ∞,

G = ∑_{k=0}^{m} ∑_{j=0}^{∞} b_{k,j}E_{k,j}, ∑_{k=0}^{m} ∑_{j=0}^{∞} |b_{k,j}|^{2} = 1.

Then

\frac{|⟨S_{z}^{∗}G, F ⟩|}{∥L(0)G∥} = \frac{|⟨ ∑_{k=0}^{m} ∑_{j=0}^{∞} b_{k,j}E_{k,j}, ∑_{k=0}^{m} ∑_{j=0}^{∞} a_{k,j}S_{z}E_{k,j}⟩|}{∥ ∑_{k=0}^{m} ∑_{j=0}^{∞} b_{k,j} \frac{λ_{k}(w)φ^{j}(w)}{\sqrt{j+1}} ∥}

= \frac{| ∑_{k=0}^{m} ⟨ ∑_{j=0}^{∞} b_{k.j}E_{k,j}, ∑_{j=0}^{∞} a_{k,j}S_{z}E_{k,j}⟩|}{\sqrt{∑{m \atop k=0} ∑{∞ \atop j=0} \frac{|b_{k,j}|^{2}}{j+1}}}

= \frac{| ∑_{k=0}^{m} ∑_{j=0}^{∞} \begin{matrix}√ & \overline{j+1} \\ √ & \overline{j+2} \end{matrix}b_{k,j+1}a_{k,j}|}{\sqrt{∑{m \atop k=0} ∑{∞ \atop j=0} \frac{|b_{k,j}|^{2}}{j+1}}}

and

sup_{G∈N_{φ},∥G∥=1} \frac{|⟨S_{z}^{∗}G, F ⟩|}{∥L(0)G∥} = \sqrt{∑_{k=0}^{m} ∑_{j=0}^{∞}(j + 1)|a_{k,j}|^{2}} .

Write c_{k,j} = \sqrt{j + 1}a_{k,j}, then we have F ∈ D_{z}(M_{φ} ⊖ zM_{φ}) if and only if

F = ∑_{k=0}^{m} ∑_{j=0}^{∞} \frac{c_{k,j}E_{k,j}}{\sqrt{j + 1}} , ∑_{k=0}^{m} ∑_{j=0}^{∞} |c_{k,j}|^{2} < ∞.

So

U(R(D_{z})) = (H^{2}(Γ) ⊖ φH^{2}(Γ)) ⊗ H^{2}(Γ).

The above fact also can be proved using (5.1) and (5.2).

It follows directly from (5.3) that S_{z} on N_{φ} is essentially normal if and only if φ is a finite Blaschke product. Now we take a look at the essential normality of S_{w}. Some facts about the space H^{2}(Γ) ⊖ φH^{2}(Γ) need to be mentioned here. We recall that the Jordan block S(φ) is defined by

S(φ)g = P_{φ}wg, g ∈ H^{2}(Γ) ⊖ φH^{2}(Γ),

where P_{φ} is the orthogonal projection from H^{2}(Γ) onto H^{2}(Γ) ⊖ φH^{2}(Γ). The two functions P_{φ}1 and P_{φ}wφ play important roles here, and we let the operator T_{0} on H^{2}(Γ) ⊖ φH^{2}(Γ) be defined by T_{0}g = ⟨g, P_{φ}wφ⟩P_{φ}1. One verifies that

T_{0}^{∗}T_{0}g = ∥P_{φ}1∥^{2}⟨g, P_{φ}wφ⟩P_{φ}wφ, T_{0}T_{0}^{∗}g = ∥P_{φ}wφ∥^{2}⟨g, P_{φ}1⟩P_{φ}1,

and

(5.4) I − S(φ)^{∗}S(φ) = ∥P_{φ}1∥^{−2}T_{0}^{∗}T_{0}, I − S(φ)S(φ)^{∗} = ∥P_{φ}wφ∥^{−2}T_{0}T_{0}^{∗}.

For every g(w) ∈ H^{2}(Γ_{w}) ⊖ φH^{2}(Γ_{w}), we decompose wg as

wg(w) = S(φ)g(w) + (I − P_{φ})wg(w).

Using the facts that (I − P_{φ})wg = ⟨wg, φ⟩φ, P_{φ}1 = 1 − φ(0)φ and S_{φ} = S_{z}, where S_{φ}g = P_{N_{φ}}φg, we have

S_{w}g(w)e_{j}(z, φ(w))

= ∑_{m,n}⟨wg(w)e_{j}(z, φ(w)), E_{m,n}⟩E_{m,n}

= ∑_{m,n} ⟨(S(φ)g)e_{j}(z, φ(w)) + ⟨wg, φ⟩\frac{φP_{φ}1}{1 − φ(0)φ}e_{j}(z, φ(w)), E_{m,n}⟩E_{m,n}

= (S(φ)g)e_{j}(z, φ(w)) + ⟨wg, φ⟩ ∑_{m,n} ⟨\frac{φP_{φ}1}{1 − φ(0)φ}e_{j}(z, φ(w)), E_{m,n}⟩E_{m,n}

= (S(φ)g)e_{j}(z, φ(w)) + ⟨g, P_{φ}wφ⟩(I − φ(0)S_{z})^{−1}S_{z}(P_{φ}1 · e_{j}(z, φ(w))).

So

(5.5)  US_{w}U^{∗} = S(φ) ⊗ I + T_{0} ⊗ (I − φ(0)B)^{−1}B.

For further discussion, we assume φ is not a singular inner function, i.e., φ has a zero in 𝔻. We first look at the case when φ(0) = 0. In this case (5.5) reduces to the cleaner expression

(5.6)  US_{w}U^{∗} = S(φ) ⊗ I + T_{0} ⊗ B.

Using (5.6) and the fact S(φ)^{∗}T_{0} = T_{0}S(φ)^{∗} = 0, one easily verifies that

US_{w}^{∗}S_{w}U^{∗} = S(φ)^{∗}S(φ) ⊗ I + T_{0}^{∗}T_{0} ⊗ B^{∗}B,

and

US_{w}S_{w}^{∗}U^{∗} = S(φ)S(φ)^{∗} ⊗ I + T_{0}T_{0}^{∗} ⊗ BB^{∗}.

Then by (5.4)

(5.7)  U[S_{w}^{∗}, S_{w}]U^{∗} = (I − S(φ)S(φ)^{∗}) ⊗ I − (I − S(φ)^{∗}S(φ)) ⊗ I
+ T_{0}^{∗}T_{0} ⊗ B^{∗}B − T_{0}T_{0}^{∗} ⊗ BB^{∗}
= T_{0}T_{0}^{∗} ⊗ (I − BB^{∗}) − T_{0}^{∗}T_{0} ⊗ (I − B^{∗}B).

Since T_{0} is of rank 1 and it is well-known that I − BB^{∗} and I − BB^{∗} are Hilbert–Schmidt, (5.7) implies that [S_{w}^{∗}, S_{w}] is Hilbert–Schmidt. The Hilbert–Schmidt norm of [S_{w}^{∗}, S_{w}] can be readily calculated in this case. First of all, P_{N_{φ}}1 = 1 and P_{N_{φ}}wφ = wφ. Let λ_{k}(w), k = 0, 1, 2, …, be an orthonormal basis of H^{2}(Γ_{w}) ⊖ φH^{2}(Γ_{w}) and λ_{0}(w) = 1. Then by (5.7),

[S_{w}^{∗},S_{w}]λ_{k}(w)e_{j}(z, φ(w))

= \frac{(T_{0}T_{0}^{∗}λ_{k}(w))e_{j}(z, φ(w))}{j + 1} − \frac{(T_{0}^{∗}T_{0}λ_{k}(w))e_{j}(z, φ(w))}{j + 2}

= \frac{λ_{k}(0)e_{j}(z, φ(w))}{j + 1} − \frac{⟨λ_{k}(w), wφ(w)⟩wφ(w)e_{j}(z, φ(w))}{j + 2},

and one calculates that

∑_{k} ∥[S_{w}^{∗}, S_{w}]λ_{k}(w)e_{j}(z, φ(w))∥^{2} = \frac{1}{(j + 1)^{2}} + \frac{1}{(j + 2)^{2}} − \frac{2|φ′(0)|^{2}}{(j + 1)(j + 2)},

from which it follows that

∥[S_{w}^{∗}, S_{w}]∥_{H.S}^{2} = \frac{π^{2}}{3} − 1 − 2|φ′(0)|^{2}.

In the case φ(0) ≠ 0, we need an additional general fact. For α ∈ 𝔻, we let τ_{α}(w) = \frac{α−w}{1−αw}. So if we let operator U_{α} be defined by

U_{α}(f)(z, w) := \frac{\sqrt{1 − |α|^{2}}}{1 − αw} f(z, τ_{α}(w)), f ∈ H^{2}(𝔻^{2}),

then it is well-known that U_{α} is a unitary. We let M′ = U_{α}([z − φ]) = [z − φ(τ_{α})] and N′ = H^{2}(𝔻^{2}) ⊖ M′. The two variable Jordan block on N′ is denoted by (S′_{z}, S′_{w}). Then by [25],

U_{α}S_{z}U_{α}^{∗} = S′_{z}, U_{α}S_{w}U_{α}^{∗} = τ_{α}(S′_{w}).

Since τ_{α}(τ_{α}(w)) = w, we also have

U_{α}τ_{α}(S_{w})U_{α}^{∗} = S′_{w}.

So if φ(0) ≠ 0, we pick any zero of φ, say α. Since φ(τ_{a}(0)) = φ(α) = 0, [S′_{w}^{∗}, S′_{w}] is Hilbert–Schmidt by the above calculations, and it then follows that [S_{w}^{∗}, S_{w}] is Hilbert–Schmidt (cf. [20, Lemma 1.3]). So in conclusion, when φ is not singular [S_{w}^{∗}, S_{w}] is Hilbert–Schmidt on N_{φ}.

These calculations on S_{z} and S_{w} prove the following theorem.

Theorem 5.3. Let φ be an one variable inner function. Then N_{φ} is essentially reductive if and only if φ is a finite Blaschke product.

On N_{φ}, the commutater [S_{z}^{∗}, S_{w}] can also be easily calculated. One sees that

US_{z}^{∗}S_{w}U^{∗} = (I ⊗ B^{∗}) (S(φ) ⊗ I + T_{0} ⊗ (I − φ(0)B)^{−1}B)
= S(φ) ⊗ B^{∗} + T_{0} ⊗ B^{∗}(I − φ(0)B)^{−1}B,

and

US_{w}S_{z}^{∗}U^{∗} = (S(φ) ⊗ I + T_{0} ⊗ (I − φ(0)B)^{−1}B) (I ⊗ B^{∗})
= S(φ) ⊗ B^{∗} + T_{0} ⊗ (I − φ(0)B)^{−1}BB^{∗}.

So

U[S_{z}^{∗}, S_{w}]U^{∗} = T_{0} ⊗ [B^{∗}, (I − φ(0)B)^{−1}B].

It was shown in [26] that

(5.8)  tr[f(B)^{∗}, g(B)] = ∫_{𝔻} f′(w)g′(w)dA,

456 K. Izuchi and R. Yang

where f and g are analytic functions on 𝔻 that are continuous on \overline{𝔻} and the derivatives f′ and g′ are in L_{a}^{2}(𝔻). Using (5.8), one easily verifies that [B^{∗}, (1 − φ(0)B)^{−1}B] is trace class with tr[B^{∗}, (1 − φ(0)B)^{−1}B] = 1. Therefore, [S_{z}^{∗}, S_{w}] is trace class with

tr[S_{z}^{∗}, S_{w}] = tr T_{0} · tr[B^{∗}, (I − φ(0)B)^{−1}B]
= tr T_{0}
= φ′(0).

Example 2. As we have remarked before that S_{z} on N_{w} is equivalent to the Bergman shift B and S_{z} = S_{w} in this case, and moreover φ′ = 1. So from the calculations above

tr[B^{∗}, B] = 1, and ∥[B^{∗}, B]∥_{H.S.}^{2} = \frac{π^{2}}{3} − 3.

References

[1] Bercovici, Hari. Operator theory and arithmetic in H^{∞}. Mathematical Surveys and Monographs, 26. American Mathematical Society, Providence, RI, 1988. xii+275 pp. ISBN: 0-8218-1528-8. MR0954383 (90e:47001), Zbl 0653.47004.

[2] Cima, Joseph A.; Ross, William T. The backward shift on the Hardy space. Mathematical Surveys and Monographs, 79. American Mathematical Society, Providence, RI, 2000. xii+199 pp. ISBN: 0-8218-2083-4. MR1761913 (2002f:47068), Zbl 0952.47029.

[3] Chen, Xiaoman; Guo, Kunyu. Analytic Hilbert modules. Chapman & Hall/CRC Research Notes in Mathematics, 433. Chapman & Hall/CRC, Boca Raton, FL, 2003. viii+201 pp. ISBN: 1-58488-399-5. MR1988884 (2004d:47024), Zbl 1048.46005.

[4] Douglas, Ronald G.; Yang, Rongwei. Operator theory in the Hardy space over the bidisk. I. Integral Equations Operator Theory 38(2000) 207–221. MR1791052 (2002m:47006), Zbl 0970.47016.

[5] Garnett, John B. Bounded analytic functions. Pure and Applied Mathematics, 96. Academic Press, New York, 1981. xvi+467 pp. ISBN: 0-12-276150-2. MR0628971 (83g:30037), Zbl 0469.30024.

[6] Guo, Kunyu. Characteristic spaces and rigidity for analytic Hilbert modules. J. Funct. Anal. 163 (1999) 133–151. MR1682835 (2000b:46090), Zbl 0937.46047.

[7] Guo, Kunyu. Algebraic reduction for Hardy submodules over polydisk algebras. J. Operator Theory 41 (1999) 127–138. MR1675180 (2000b:46091), Zbl 0990.46033.

[8] Guo, Kunyu. Equivalence of Hardy submodules generated by polynomials. J. Funct. Anal. 178 (2000) 343–371. MR1802898 (2002f:47128), Zbl 0977.46028.

[9] Guo, Kunyu. Podal subspaces on the unit polydisk. Studia Math. 149 (2002) 109– 120. MR1881248 (2002m:46082), Zbl 1018.46028.

[10] Guo, Kunyu; Yang, Rongwei. The core function of submodules over the bidisk. Indiana Univ. Math. J. 53 (2004) 205–222. MR2048190 (2005m:46048), Zbl 1062.47009.

[11] Hoffman, Kenneth. Banach spaces of analytic functions. Prentice-Hall Series in Modern Analysis. Prentice-Hall Englewood Cliffs, NJ, 1962. xiii+217 pp. MR0133008 (24 #A2844), Zbl 0117.34001.

[12] Izuchi, Keiji; Nakazi, Takahiko; Seto, Michio. Backward shift invariant subspaces in the bidisc. II. J. Operator Theory 51 (2004) 361–376. MR2074186 (2005c:47008), Zbl 1055.47009.

[13] Izuchi, Keiji; Nakazi, Takahiko; Seto, Michio. Backward shift invariant subspaces in the bidisc. III. Acta Sci. Math. (Szeged) 70 (2004) 727–749. MR2107538 (2005i:47013).

[14] Izuchi, Keiji; Yang, Rongwei. Strictly contractive compression on backward shift invariant subspaces over the torus. Acta Sci. Math. (Szeged) 70 (2004) 147–165. MR2072696 (2005e:47019), Zbl 1062.47017.

[15] Mandrekar, V. The validity of Beurling theorems in polydiscs. Proc. Amer. Math. Soc. 103 (1988) 145–148. MR0938659 (90c:32008), Zbl 0658.47033.

[16] Nakazi, Takahiko. An outer function and several important functions in two variables. Arch. Math. (Basel) 66 (1996) 490–498. MR1388099 (97d:32004), Zbl 0856.32002.

[17] Rudin, Walter. Function theory in polydiscs. Benjamin, New York, 1969. vii+188 pp. MR0255841 (41 #501), Zbl 0177.34101.

[18] Sz.-Nagy, Béla; Foias, Ciprian. Harmonic analysis of operators on Hilbert space. Translated from the French and revised. North-Holland, Amsterdam; American Elsevier, New York; Akad. Kiadó, Budapest, 1970. xiii+389 pp. MR0275190 (43 #947).

[19] Stessin, Michael; Zhu, Kehe. Joint composition operators in several complex variables. Preprint.

[20] Yang, Rongwei. The Berger–Shaw theorem in the Hardy module over the bidisk. J. Operator Theory 42 (1999) 379–404. MR1717024 (2000h:47040), Zbl 0991.47015.

[21] Yang, Rongwei. Operator theory in the Hardy space over the bidisk. III. J. Funct. Anal. 186 (2001) 521–545. MR1864831 (2002m:47008), Zbl 1049.47501.

[22] Yang, Rongwei. Operator theory in the Hardy space over the bidisk. II. Integral Equations Operator Theory 42 (2002) 99–124. MR1866878 (2002m:47007), Zbl 1002.47012.

[23] Yang, Rongwei. On two-variable Jordan blocks. Acta Sci. Math. (Szeged) 69 (2003) 739–754. MR2034205 (2004j:47011), Zbl 1052.47004.

[24] Yang, Rongwei. Beurling’s phenomenon in two variables. Integral Equations Operator Theory 48 (2004) 411–423. MR2038510 (2004j:46038), Zbl 1061.46023.

[25] Yang, Rongwei. On two variable Jordan block. II. Integral Equations Operator Theory 56 (2006) 431–449. MR2270846 (2007i:47006).

[26] Zhu, Kehe. A trace formula for multiplication operators on invariant subspaces of the Bergman space. Integral Equations Operator Theory 40 (2001) 244–255. MR1831829 (2002c:47074), Zbl 0995.47017.

Department of Mathematics, Niigata University, Niigata, 950-2181, Japan izuchi@m.sc.niigata-u.ac.jp

Department of Mathematics and Statistics, SUNY at Albany, Albany, NY 12047, U.S.A. ryang@math.albany.edu

This paper is available via http://nyjm.albany.edu/j/2008/14-21.html.