Inner vectors for Toeplitz operators

For $\varphi\in H^{\infty}$, the bounded analytic functions on the open unit disk $\mathbb{D}$, let

denote the analytic Toeplitz operator on the classical Hardy space $H^{2}$. In this paper we survey, continue, and synthesize some discussions begun in [4, 10, 11] dealing with the notion of an “inner vector” for $T_{\varphi}$ along with the general notion of an inner vector for a contraction on a Hilbert space. We connect these results with the operator-valued Poisson kernel and some work from [2, 3] concerning “factoring an $L^{1}$ function through a contraction”. Along the way we also produce some interesting examples and reformulations of these connections.

We begin with the definition of an inner vector for a Toeplitz operator from [10]. Recall that the inner product on the Hardy space $H^{2}$ is

where $m$ is normalized Lebesgue measure on the unit circle $\mathbb{T}$. As is tradition, we equate an $f\in H^{2}$ with its $L^{2}=L^{2}(\mathbb{T},m)$ radial boundary function, i.e.,

for almost every $\zeta\in\mathbb{T}$. We will also use the term inner function (without any qualifiers like in Definition 2.2 below) to denote an $H^{\infty}$ function that has unimodular boundary values almost everywhere. Classical theory [6] says that an inner function $I$ can be factored uniquely as $I=\xi BS_{\mu}$, where $\xi$ is a unimodular constant, $B$ is a Blaschke product, and $S_{\mu}$ is a singular inner function associated with a positive measure $\mu$ on $\mathbb{T}$ that is singular with respect to $m$. We say the degree of $I$ is equal to $d$ if $I$ is a finite Blaschke product of order $d$, and equal to infinity otherwise. Furthermore, any function $f\in H^{2}$ can be factored, uniquely up to multiplicative constants, as $f=IG$, where $I$ is an inner function and $G\in H^{2}$ is an outer function.

For $\varphi\in H^{\infty}$ the analytic Toeplitz operator $T_{\varphi}$ from (1.1) is a bounded operator on $H^{2}$ whose norm $\|T_{\varphi}\|$ satisfies

Also recall that the adjoint $T_{\varphi}^{*}$ of $T_{\varphi}$ satisfies $T_{\varphi}^{*}=T_{\overline{\varphi}}$, where $T_{\overline{\varphi}}f=P(\overline{\varphi}f)$ and $P$ is the Riesz projection of $L^{2}$ onto $H^{2}$. When $\varphi$ is an inner function, observe from (2.1) that $T_{\varphi}$ is an isometry. See [8, Ch. 4] for the details of these basic Toeplitz operator facts and [1] for a definitive treatise.

When $\varphi(z)=z$, one can see from Fourier analysis that the $T_{z}$-inner vectors are precisely the inner functions. Also observe that replacing $\varphi$ with $c\varphi$, where $c>0$, in Definition 2.2 does not change whether or not a function $f$ is $T_{\varphi}$-inner. Thus we can always assume, by scaling $\varphi$, that

the closed unit ball of $H^{\infty}$. This normalization will be important when we need $T_{\varphi}$ to be a contraction operator since in this case $\|T_{\varphi}\|=\|\varphi\|_{\infty}\leqslant 1$. Immediate from Definition 2.2 and the inner product formula from (2.1) are the following facts.

If $u$ denotes the inner factor of $\varphi$, it is known [8, p. 108] that

the model space corresponding to $u$. Thus we have the simple corollary.

This corollary gives us many specific examples of $T_{\varphi}$-inner vectors. For example, if $\lambda\in\mathbb{D}$, the reproducing kernel functions

belong to $\mathcal{K}_{u}$. In fact, finite linear combinations of these functions are dense in $\mathcal{K}_{u}$ [8, Ch. 5]. Since

then

are $T_{\varphi}$-inner functions.

When $\varphi=u$ is a finite Blaschke product, then the model space $\mathcal{K}_{u}$ is a certain finite dimensional space of rational functions that are analytic in a neighborhood of $\overline{\mathbb{D}}$ [8, p. 117]. Furthermore, as we will see in a moment in Theorem 3.12, every $T_{u}$-inner function is bounded. However, when $u$ is not a finite Blaschke product then $\mathcal{K}_{u}$ is infinite dimensional [8, p. 117] and, since multiplication by an inner function $I$ is an isometry on $H^{2}$ (see (2.1)), $I\mathcal{K}_{u}$ is a closed infinite dimensional subspace of $L^{2}$. By a theorem of Grothendieck, it will contain an unbounded function. Putting this all together, we obtain the following.

A specific version of this was pointed out in [10, p. 103].

Of course one needs to discuss the case when $\varphi$ is an outer function. Since $\varphi H^{2}$ is dense in $H^{2}$ [8, p. 86], we see that $\ker T_{\overline{\varphi}}=\{0\}$. In this case, it is not clear that there are any $T_{\varphi}$-inner functions. Indeed, we do not see any obvious ones like $I\ker T_{\overline{\varphi}}$ since, in this case, $\ker T_{\overline{\varphi}}=\{0\}$.

Using some ideas from [10], we can use the Wold decomposition [9] to explore the inner vectors for certain Toeplitz operators. Observe that when $u$ is an inner function the Toeplitz operator $T_{u}$ is an isometry on $H^{2}$. Thus the Wold decomposition of $H^{2}$ with respect to $T_{u}$ becomes

where

Thus

The above decomposition says that every $f\in H^{2}$ has a unique expansion as

Furthermore, for each integer $N\geqslant 1$,

This leads us to the following.

Though this is just a restatement of the condition for $f$ to be $T_{u}$-inner, it is useful for producing more tangible examples of $T_{u}$-inner functions.

From Corollary 2.4 we know, for an inner function $I$, that any unit vector from the set $\{I\ker T_{\overline{u}}:\mbox{$I$ is inner}\}$ is a $T_{\overline{u}}$-inner vector. Perhaps one might think we have equality here. Indeed, sometimes we do. For example, if $u(z)=z$, then $\ker T_{\overline{z}}=\mathbb{C}$ and, as discussed earlier, the $T_{z}$-inner vectors are precisely the inner functions. Here is another positive example of when the unit vectors from $\{I\ker T_{\overline{u}}:\mbox{$I$ is inner}\}$ constitute the complete set of $T_{u}$-inner vectors.

However, in general, the unit vectors from $\{I\ker T_{\overline{u}}:\mbox{$I$ is inner}\}$ form a proper subset of the $T_{u}$-inner vectors. One can see this with the following example.

The papers [10, 11] yield a description of the $T_{u}$-inner vectors. From the Wold decomposition (3.1) we see that any $f\in H^{2}$ can be written as

If $\{v_{j}\}_{j\geqslant 1}$ is an orthonormal basis for $\mathcal{K}_{u}$, then we can expand things a bit further and write

Observe that

and that

Thus for each $j$, $\sum_{k\geqslant 0}|c_{j,k}|^{2}<\infty$ and so

defines a function in $H^{2}$ (square summable power series). By the Littlewood subordination principle [8, p. 126], $f_{j}\circ u$ also belongs to $H^{2}$.

Thus every unit vector $f\in H^{2}$ has the unique representation

where $f_{j}\in H^{2}$ with $\sum_{j\geqslant 1}\|f_{j}\|^{2}<\infty$, and $\{v_{j}\}_{j\geqslant 1}$ is an orthonormal basis for $\mathcal{K}_{u}$. Furthermore, as observed in [10, Prop. 1] (and can be proved using the above calculation), if

as in (3.9), then

When $u$ is a finite Blaschke product, then $\mathcal{K}_{u}$ is finite dimensional. In this case (3.9) is finite and each basis vector $v_{j}$ is a rational function that is analytic in a neighborhood of $\overline{\mathbb{D}}$ [8, Ch. 5]. From here it follows that every $T_{u}$-inner vector is a bounded function. Contrast this with Corollary 2.5 which says that when $u$ is not a finite Blaschke product there are always $T_{u}$-inner vectors that are unbounded functions.

The two papers [10, 11] go further and discuss an “inner-outer” factorization of any $f\in H^{2}$ in terms of $T_{u}$-inner and $T_{u}$-outer vectors. They also discuss the concept of $T_{u}$-inner in $H^{p}$, for $p>1$, along with some properties of the norms of $T_{u}$-inner vectors as well as their growth near $\mathbb{T}$.

We can rephrase the language of inner vectors for Toeplitz operators in terms of operator-valued Poisson kernels [2]. Moreover, using this new language, we can extend our discussion to inner vectors for contractions on Hilbert spaces. For $\lambda\in\mathbb{D}$ and $\xi\in\mathbb{T}$, define

and observe that this can be written as

which is the standard Poisson kernel. Classical theory says that for any $g\in L^{1}=L^{1}(\mathbb{T},m)$ the function

is harmonic on $\mathbb{D}$ with

for almost every $\zeta\in\mathbb{T}$. Furthermore, if $\mu$ is a finite complex measure on $\mathbb{T}$, we have

where

are the Fourier coefficients of $\mu$. We will now discuss an operator version of the Poisson kernel.

For a contraction $T$ on a Hilbert space $\mathcal{H}$, we imitate the formula in (4.1) and define, for $\lambda\in\mathbb{D}$, the operator-valued Poisson kernel $K_{\lambda}(T)$ as

By the spectral radius formula, notice how $\sigma(T)\subseteq\overline{\mathbb{D}}$ and thus the formula for $K_{\lambda}(T)$ above makes sense. A computation with Neumann series will show that for $r\in[0,1)$ and $\theta\in[0,2\pi)$

The operator identity

from [2, Lemma 2.4] shows that for each $\mathbf{x}\in\mathcal{H}$

Moreover, the function

is harmonic on $\mathbb{D}$. Hence, a classical harmonic analysis result of Herglotz ([6, p. 10] or [8, p. 17]) produces a unique positive finite Borel measure $\mu_{T,\mathbf{x}}$ on $\mathbb{T}$ such that

Since $K_{0}(T)=I$ we have

and so $\mu_{T,\mathbf{x}}$ is a probability measure.

As we defined for Toeplitz operators earlier in Definition 2.2, we say that a unit vector $\mathbf{x}$ is $T$-inner if

Note that $\mathbf{x}$ is $T$-inner if and only if $\mathbf{x}$ is $T^{*}$-inner. From (4.4) we see that $\mathbf{x}$ is $T$-inner if and only if $\langle K_{\lambda}(T)\mathbf{x},\mathbf{x}\rangle=1$ for all $\lambda\in\mathbb{D}$, or equivalently,

By (4.3) this is equivalent to the condition $\mu_{T,\mathbf{x}}=m$. This gives us the following.

For an inner function $u$, note that $T_{u}$ is an isometry, hence a contraction. Thus we can apply the above analysis to $\mu_{T_{u},f}$.

Notice how this gives us another way of thinking about Theorem 3.12: a unit vector $f\in H^{2}$ is $T_{u}$-inner if and only if $\mu_{T_{u},f}=m$.

This brings us to an interesting related question. One can also show that for any $f,g\in H^{2}$, we can define the harmonic function $\langle K_{\lambda}(T_{u})f,g\rangle$ on $\mathbb{D}$ and prove this function also has bounded integral means. This yields, via Herglotz’s theorem, a complex valued measure $\mu_{T_{u},f,g}$ on $\mathbb{T}$ for which

See [2, Prop. 2.6] for details. A similar calculation used to prove Proposition 4.6 shows that

In the above formula, $f_{j}$ and $g_{j}$ come from the representations of $f$ and $g$ from (3.10). A general result from [3] says that given any $F\in L^{1}$ and a non-constant inner function $u$ that is not an automorphism, there are $f,g\in H^{2}$ for which

$m$-almost everywhere. In the language of [3] this says that any $F\in L^{1}$ can be “factored through $T_{u}$”. Equivalently stated, using (4.10) and (4.11), we have

This is an interesting representation for $L^{1}$ functions and a refinement of the one from [3].

For any fixed $\alpha\in\mathbb{T}$ and inner function $u$, the function

is a positive harmonic function on $\mathbb{D}$. Thus by Herglotz’s theorem, there is a unique positive measure $\sigma_{\alpha}$ on $\mathbb{T}$ for which

The family of measures $\{\sigma_{\alpha}:\alpha\in\mathbb{T}\}$ is called the family of Clark measures corresponding to $u$. Let us record some important facts about this family of measures. Proofs can be found in [5].

First, one can use the fact that $u$ is an inner function, along with standard harmonic analysis, to prove that each $\sigma_{\alpha}$ is singular with respect to $m$. Second, if $E_{\alpha}$ is defined to be the set of $\xi\in\mathbb{T}$ for which

then $E_{\alpha}$ is a Borel subset of $\mathbb{T}$ with

In other words, $\sigma_{\alpha}$ is “carried” by $E_{\alpha}$. From this we also see that the measures $\{\sigma_{\alpha}:\alpha\in\mathbb{T}\}$ are singular with respect to each other. Third, a beautiful disintegration theorem of Aleksandrov says that if $g\in L^{1}$ then for $m$-almost every $\alpha\in\mathbb{T}$, integral

is well defined. Moreover this almost everywhere defined function

is integrable with respect to $m$ and

Using Clark measures, we can use a technique from [11] to compute a formula for $\langle K_{\lambda}(T_{u})f,f\rangle$ along with the measure $d\mu_{T_{u},f}/dm$. This gives us another way to think about the formula (4.11). The result here is the following.