2.1 Multipliers of Fock space

Left multiplications by the d independent NC variables $\mathfrak {z} = ( \mathfrak {z} _1 , \ldots , \mathfrak {z} _d )$ define isometries on the Fock space with pairwise orthogonal ranges,

$$ \begin{align*}L_k := M^L _{\mathfrak{z} _k}, \quad \quad L_j ^* L_k = \delta _{j,k} I.\end{align*} $$

It follows that the row d-tuple $L := (L_1 , \ldots , L_d ) : \mathbb {H} ^2 _d \otimes \mathbb {C} ^d \rightarrow \mathbb {H} ^2 _d$ is an isometry from several copies of $\mathbb {H} ^2 _d$ into itself. Such an isometry is called a row isometry, and we call this row isometry of left multiplications on the Fock space the left free shift and its components the left free shifts. Similarly, one can define the right free shifts, $R_k := M^R _{\mathfrak {z} _k}$ , as right multiplication by the independent NC variables as well as the row isometric right free shift, $R = (R _1 , \ldots , R _d)$ . The letter reversal map $\mathrm {t} : \mathbb {F} ^d \rightarrow \mathbb {F} ^d$ , which reverses the order of letters in any word $\omega \in \mathbb {F} ^d$ , defines an involution on the free monoid,

$$ \begin{align*}\omega = i_1 \cdots i_n \, \mapsto \, \omega ^{\mathrm{t}} := i_n \cdots i_1.\end{align*} $$

Given a word, $\omega = i_1 \cdots i_n \in \mathbb {F} ^d$ , the length of $\omega $ is $|\omega | =n$ and $| \emptyset | := 0$ . The free monomials $ \{ e_\omega := \mathfrak {z} ^\omega | \ \omega \in \mathbb {F} ^d \}$ define a standard orthonormal basis of $\mathbb {H} ^2 _d \simeq \ell ^2 (\mathbb {F} ^d )$ , and the letter reversal map gives rise to a unitary involution of the Fock space, $U_{\mathrm {t}}$ , defined by $U_{\mathrm {t}} \mathfrak {z} ^\omega = \mathfrak {z} ^{\omega ^{\mathrm {t}}}$ . Here, $e_\emptyset = \mathfrak {z} ^\emptyset =: 1$ is called the vacuum vector of the Fock space. It is straightforward to verify that $U_{\mathrm {t}} L_k U_{\mathrm {t}} = R_k$ , so that the left shifts are isomorphic to the right shifts.

The NC Hardy algebra, $\mathbb {H} ^\infty _d$ , of uniformly bounded NC functions can be identified, completely isometrically, with the unital Banach algebra of left multipliers of the NC Hardy space, $\mathbb {H} ^2 _d$ [Reference Salomon, Shalit and Shamovich62, Theorem 3.1]. That is, given any NC function, $F \in \mathbb {H} ^\infty _d$ , and $h \in \mathbb {H} ^2 _d$ , the left multiplication operator $M^L _F : \mathbb {H} ^2 _d \rightarrow \mathbb {H} ^2 _d$ , defined by

$$ \begin{align*}h(Z) \mapsto F(Z) \cdot h(Z),\end{align*} $$

is bounded and $\| M ^L _F \| = \| F \| _\infty $ , where $\| \cdot \| _\infty $ denotes the supremum norm over $\mathbb {B} ^d _{\mathbb {N}}$ . For any free polynomial $p \in \mathbb {C} \{ \mathfrak {z} \} := \mathbb {C} \{ \mathfrak {z} _1 , \ldots , \mathfrak {z} _d \}$ , one can check that $p(L) = M^L _p$ , and so we employ the notation $F(L) := M^L _F$ . Similarly, if $p \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ , then, $M^R _p = p ^{\mathrm {t}} (R) = U_{\mathrm {t}} p(L) U_{\mathrm {t}}$ , acts as right multiplication by p, where if h is a formal power series, $h(\mathfrak {z} ) = \sum \hat {h} _\omega \mathfrak {z} ^\omega $ ,

$$ \begin{align*}h^{\mathrm{t}} (\mathfrak{z} ) := \sum \hat{h} _\omega \mathfrak{z} ^{\omega ^{\mathrm{t}}} = \sum \hat{h} _{\omega ^{\mathrm{t}}} \mathfrak{z} ^\omega.\end{align*} $$

In particular, if $h \in \mathbb {H} ^2 _d$ , $h^{\mathrm {t}} = U_{\mathrm {t}} h$ . The left and right multiplier algebras of $\mathbb {H} ^2 _d$ are unitarily equivalent via the unitary letter reversal involution $U_{\mathrm {t}}$ and can be identified with the left and right analytic Toeplitz algebras, $\mathscr {L} ^\infty _d := \mathrm {Alg} \{ I , L_1 , \ldots , L_d \} ^{-WOT}$ and $\mathscr {R} ^\infty _d =\{ I , R_1 , \ldots , R_d \} ^{-WOT} = U_{\mathrm {t}} \mathscr {L} ^\infty _d U_{\mathrm {t}}$ , where $WOT$ denotes the weak operator topology. Since $p (R) = M^R _{p ^{\mathrm {t}}}$ for any $p \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ , we will write $G (R) = M^R _{G ^{\mathrm {t}}}$ for any $G \in \mathbb {H} ^\infty _d$ . Namely, $G \in \mathbb {H} ^\infty _d$ if and only if $G^{\mathrm {t}} \in \mathbb {H} ^{\infty; \mathrm {t}} _d := \mathrm {t} \circ \mathbb {H} ^\infty _d$ . We will use the following terminology: A left or right multiplier is inner if it is isometric and outer if it has dense range.

2.2 Noncommutative reproducing kernel Hilbert spaces

In syzygy with classical Hardy space theory, the Fock space is a (noncommutative) reproducing kernel Hilbert space, in the sense that for any $Z \in \mathbb {B} ^d _n$ and vectors $y,v \in \mathbb {C} ^n$ , the matrix-entry point evaluation, $\ell _{Z,y,v} : \mathbb {H} ^2 _d \rightarrow \mathbb {C}$ ,

$$ \begin{align*}h \mapsto y^* h(Z) v,\end{align*} $$

is a bounded linear functional. Equivalently the linear map $h \mapsto h(Z)$ is bounded as a map from $\mathbb {H} ^2 _d$ into the Hilbert space, $\mathbb {C} ^{n \times n}$ , equipped with the Hilbert–Schmidt inner product. By the Riesz lemma, $\ell _{Z,y,v}$ is implemented by inner products against vectors $K \{ Z , y , v \} \in \mathbb {H} ^2 _d$ , which we call NC Szegö kernel vectors.

In greater generality, let $\mathbb {C} ^d _{\mathbb {N}} := \bigsqcup _{n=1} ^\infty \mathbb {C} ^d _n$ denote the d-dimensional complex NC universe. Here, recall that we define $\mathbb {C} ^d _n := \mathbb {C} ^{n\times n} \otimes \mathbb {C} ^{1\times d}.$ A subset $\Omega \subseteq \mathbb {C} ^d _{\mathbb {N}}$ is an NC set if it is closed under direct sums, and we write $\Omega = \bigsqcup \Omega _n$ where $\Omega _n := \Omega \bigcap \mathbb {C} ^d _n$ . A Hilbert space, $\mathcal {H}$ , of free noncommutative functions on $\Omega $ is a noncommutative reproducing kernel Hilbert space (NC-RKHS), if for any $n\in \mathbb {N}$ , $Z \in \Omega _n$ and $y,v \in \mathbb {C} ^n$ , the linear point evaluation functional

$$ \begin{align*}h \mapsto y^* h(Z) v\end{align*} $$

is bounded on $\mathcal {H}$ [Reference Ball, Marx and Vinnikov8]. As before, the Riesz lemma implies that these functionals are implemented by taking inner products against point evaluation or NC kernel vectors $k \{ Z, y ,v \} \in \mathcal {H}$ . Given any such NC-RKHS, $Z \in \Omega _n$ , and $W \in \Omega _m$ , one can define a completely bounded map on $n \times m$ complex matrices by: $k(Z,W) [ \cdot ] : \mathbb {C} ^{n\times m} \rightarrow \mathbb {C} ^{n\times m}$ ,

$$ \begin{align*}y^* k(Z,W) [vu^*] x := \langle {k\{ Z,y,v \}} , {k \{ W,x ,u \} } \rangle_{\mathcal{H}},\end{align*} $$

and this map is completely positive if $Z=W$ [Reference Ball, Marx and Vinnikov8]. Here and throughout, all inner products are conjugate-linear in their first argument. Following [Reference Ball, Marx and Vinnikov8], we call $k(Z,W)[\cdot ]$ the completely positive noncommutative (CPNC) reproducing kernel of $\mathcal {H}$ , and we write $\mathcal {H} = \mathcal {H} _{nc} (k)$ . One can check that adjoints of left and right multipliers of an NC-RKHS have a familiar action on NC kernel vectors:

$$ \begin{align*}(M^L _F ) ^* k\{ Z , y ,v \} = k\{ Z , F(Z) ^* y , v \} \quad \mbox{and} \quad (M^R _G ) ^* k \{ Z , y ,v \} = k \{ Z , y , G (Z) v \}.\end{align*} $$

All NC-RKHS in this paper will be Hilbert spaces of free NC functions in the unit row-ball $\mathbb {B} ^d _{\mathbb {N}}$ ,

$$ \begin{align*}\mathbb{B} ^d _{\mathbb{N}} = \bigsqcup _{n=1} ^\infty \mathbb{B} ^d _n, \quad \quad \mathbb{B} ^d _n := \left\{ Z \in \mathbb{C} ^{n\times n} \otimes \mathbb{C} ^{1\times d} \bigg| \, ZZ^* = Z_1 Z_1 ^* + \cdots + Z_d Z_d ^* < I_n \right\}.\end{align*} $$

In the case of the Fock space, $\mathbb {H} ^2 _d = \mathcal {H} _{nc} (K)$ , where K is the NC Szegö kernel: Given $Z \in \mathbb {B} ^d _n, W \in \mathbb {B} ^d _m$ , and $P \in \mathbb {C} ^{n \times m}$ ,

$$ \begin{align*}K(Z,W) := \left( \mathrm{id} _{n,m} [ \cdot ] - \mathrm{Ad} _{Z, W^*} [\cdot ] \right) ^{-1} \circ P = \sum _{\omega \in \mathbb{F} ^d} Z^\omega P W^{*\omega},\end{align*} $$

$\mathrm {Ad} _{Z,W^*} [P] := Z_1 P W_1 ^* + \cdots + Z_d P W_d ^*$ .

2.3 NC rational functions

As described in the introduction, a complex NC rational expression is any syntactically consistent combination of the several NC variables $\mathfrak {z} _1 , \ldots , \mathfrak {z} _d$ , the complex scalars, $\mathbb {C}$ , the operations $+ , \cdot , ^{-1}$ , and parentheses $\left ( , \right )$ with domain $\mathrm {Dom} \, \mathrm {r} = \bigsqcup _{n=1} ^\infty \mathrm {Dom} _n \, \mathrm {r}$ , where

$$ \begin{align*}\mathrm{Dom} _n \, \mathrm{r} := \bigsqcup _{n=1} ^\infty \left\{ X = (X _1 , \ldots , X_d ) \in \mathbb{C} ^{n\times n} \otimes \mathbb{C} ^{1 \times d} \bigg| \ \mathrm{r} (X) \ \mbox{is defined} \right\}.\end{align*} $$

We will use the notation $\mathbb {C} ^d _n := \mathbb {C} ^{n\times n} \otimes \mathbb {C} ^{1\times d}$ for a row d-tuple of complex $n\times n$ matrices. An NC rational expression is valid, if its domain is nonempty. An NC rational function, $\mathfrak {r}$ , is then an equivalence class of valid NC rational expressions with respect to the relation $\mathrm {r} _1 \equiv \mathrm {r} _2$ if $\mathrm {r} _1 (X) = \mathrm {r} _2 (X)$ for all $X \in \mathrm {Dom} \, \mathrm {r} _1 \bigcap \mathrm {Dom} \, \mathrm {r} _2$ . (By [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov47, Footnote, p. 52], given any two valid NC rational expressions, $\mathrm {r} _k$ , their domains at level n, $\mathrm {Dom} _n \, \mathrm {r} _k = \mathrm {Dom} \, \mathrm {r} _k \cap \mathbb {C} ^d _n$ , have nontrivial intersection for sufficiently large n.) The set of all NC rational functions in d variables with coefficients in $\mathbb {C}$ is a division ring or skew field, and is, in fact, the universal skew field of fractions for the ring $\mathbb {C} \{ \mathfrak {z} \} = \mathbb {C} \{ \mathfrak {z} _1 , \ldots , \mathfrak {z} _d \}$ of complex NC polynomials [Reference Amitsur3, Reference Cohn14], [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov47, Proposition 2.2].

Any NC rational function in d-variables, $\mathfrak {r}$ , which is regular at $0$ , has a unique (up to joint similarity) minimal descriptor realization. Namely, there is a triple $(A, b, c)$ with $A \in \mathbb {C} ^d _n$ and $b,c \in \mathbb {C} ^n$ , so that for any $X \in \mathbb {C} ^d _m$ ,

$$ \begin{align*}\mathfrak{r} (X) = \left( b^* \otimes I_m \right) L_A (X) ^{-1} \left( c \otimes I_m \right), \quad \quad L_A (X) := I_n \otimes I_m - \sum A_j \otimes X_j,\end{align*} $$

and this realization is minimal in the sense that n is as small as possible. Here, $L_A (\cdot ) $ is called a (monic, affine) linear pencil, and we will employ the simplified notations

$$ \begin{align*}ZA = Z \otimes A := \sum _{j=1} ^d Z_j \otimes A_j,\end{align*} $$

for any $Z, A \in \mathbb {C} ^d _{\mathbb {N}}$ . Minimality implies that the realization is both observable,

$$ \begin{align*}\bigvee A^{*\omega} b = \mathbb{C} ^{n},\end{align*} $$

and controllable

$$ \begin{align*}\bigvee A^\omega c = \mathbb{C} ^n\end{align*} $$

(see, e.g., [Reference Helton, Mai and Speicher31, Section 3.1.2]). Minimal realizations are unique up to joint similarity [Reference Berstel and Reutenauer10, Theorem 2.4]. The domains of NC rational functions which are regular at $0$ have a convenient description:

Theorem [Reference Kaliuzhnyi-Verbovetskyi and Vinnikov46, Theorem 3.1], [Reference Volčič68, Theorem 3.10]

If $\mathfrak {r}$ is an NC rational function which is regular at $0$ with minimal realization $(A,b,c)$ , then

$$ \begin{align*}\mathrm{Dom} \, \mathfrak{r} = \bigsqcup _{n \in \mathbb{N}} \left\{ X \in \mathbb{C} ^d _n \bigg| \ \mathrm{det} \, L_A (X) \neq 0 \right\}.\end{align*} $$

Any NC rational $\mathfrak {r} \in \mathbb {H} ^2 _d$ is necessarily defined on $\mathbb {B} ^d _{\mathbb {N}}$ and hence is regular at $0$ . As proved in [Reference Jury, Martin and Shamovich44], an NC rational function belongs to the Fock space if and only if it is regular at $0$ and has minimal realization $(A,b,c)$ so that the joint spectral radius of the d-tuple A is less than 1 [Reference Jury, Martin and Shamovich44, Theorem A]. Here, if $A := (A_1 , \ldots , A_d ) : \mathbb {C} ^n \otimes \mathbb {C} ^d \rightarrow \mathbb {C} ^n$ is any row d-tuple of $n\times n$ matrices, we define the completely positive map $\mathrm {Ad} _{A, A^*} : \mathbb {C} ^{n\times n } \rightarrow \mathbb {C} ^{n \times n}$ by

$$ \begin{align*}\mathrm{Ad} _{A,A^*} (P) := A_1 P A_1 ^* + \cdots + A_d P A_d ^*.\end{align*} $$

The joint spectral radius, $\mathrm {spr} (A)$ , of A is then defined by the Beurling formula:

$$ \begin{align*}\mathrm{spr} (A) := \lim _k \sqrt[\leftroot{-2}\uproot{10}2k]{\| \mathrm{Ad} _{A, A^*} ^{(k)} (I_n) \|}.\end{align*} $$

By the multivariable Rota–Strang theorem, A is jointly similar to a strict row contraction if and only if $\mathrm {spr} (A) <1$ [Reference Popescu58, Theorem 3.8] (see also [Reference Salomon, Shalit and Shamovich62, Proposition 2.3 and Remark 2.6]). In particular, A is said to be pure if $\mathrm {Ad} ^{(k)} _{A , A^*} (I_n) \rightarrow 0$ , and a finite-dimensional row d-tuple, $A \in \mathbb {C} ^d _n$ , is pure if and only if $\mathrm {spr} (A) <1$ [Reference Salomon, Shalit and Shamovich62, Lemma 2.5]. If $\mathfrak {r} \in \mathbb {H} ^2 _d$ has minimal descriptor realization $(A,b,c)$ , then A is jointly similar to a finite strict row contraction $\overline {Z} \in \mathbb {B} ^d _n$ , where $\overline {Z}$ denotes entrywise complex conjugation, and one can verify that $\mathfrak {r} = K \{ Z , y , v \}$ is an NC Szegö kernel vector in the Fock space, where $y,v$ are the image of $\overline {b}, \overline {c}$ under the (conjugate of the) similarity and its inverse that intertwine A and $\overline {Z}$ [Reference Jury, Martin and Shamovich44, Proposition 3.2]. That is,

$$ \begin{align*}\langle {\mathfrak{r}} , {h} \rangle_{\mathbb{H} ^2} = y^* h(Z) v\end{align*} $$

and

$$ \begin{align*}K \{Z , y ,v \} (W) = \sum _{\omega \in \mathbb{F} ^d} \overline{y^* Z^\omega v} \, W^\omega, \quad \quad W \in \mathbb{B} ^d _{\mathbb{N}}.\end{align*} $$

In [Reference Jury, Martin and Shamovich44, Theorem A], we established (a more general version of) the following theorem which characterizes when an NC rational function belongs to $\mathbb {H} ^2 _d$ .

Given $A \in \mathbb {C} ^d _m$ and $Z \in \mathbb {C} ^d _n$ , consider the linear pencil

$$ \begin{align*}L_A (Z) = I_n \otimes I_m - Z \otimes A = I_n \otimes I_m - \sum _{j=1} ^d Z_j \otimes A_j.\end{align*} $$

Observe that

$$ \begin{align*}\| Z \otimes A \| \leq \| Z \| _{\mathrm{row}} \| A \| _{\mathrm{col}} := \left\| (Z_1 , \ldots , Z _d ) \right\| _{\mathscr{L} (\mathbb{C} ^n \otimes \mathbb{C} ^d , \mathbb{C} ^n )} \left\| \left ( \begin{smallmatrix} A_1 \\ \vdots \\ A_d \end{smallmatrix} \right) \right\|_{\mathscr{L} (\mathbb{C} ^m , \mathbb{C} ^m \otimes \mathbb{C} ^d )}.\end{align*} $$

Given $A \in \mathbb {C} ^d _m$ , we will also write

$$ \begin{align*}\mathrm{col} (A) := \left ( \begin{smallmatrix} A_1 \\ \vdots \\ A _d \end{smallmatrix} \right) \in \mathbb{C} ^{m\times m} \otimes \mathbb{C} ^d, \quad \mbox{so that} \quad \| A \| _{\mathrm{col}} = \| \mathrm{col} (A) \| _{\mathscr{L} (\mathbb{C} ^m , \mathbb{C} ^m \otimes \mathbb{C} ^d )}.\end{align*} $$

It follows that $Z\otimes A$ will be similar to a contraction if Z is jointly similar to a row contraction and A is jointly similar to a column contraction, and $Z \otimes A$ will further be similar to a strict contraction if, in addition, at least one of $Z, A$ is jointly similar to a strict row or column contraction, respectively. If $Z \otimes A$ is similar to a strict contraction, then $L_A (Z) ^{-1}$ can be expanded as a convergent geometric sum. A row d-tuple, $Z \in \mathbb {C} ^d _n$ , is said to be irreducible, if it has no nontrivial jointly invariant subspace, i.e., there is no nontrivial subspace which is invariant for every $Z_k$ , $1 \leq k \leq d$ . The following lemma will be useful in the sequel.

Proof The joint spectral radius of Z obeys $\mathrm {spr} (Z) \leq \| Z \| _{\mathrm{row}}$ . Consider the row d-tuple $Z^* = \mathrm {row} (Z^*) := (Z_1 ^* , \ldots , Z_d ^* )$ . Then,

$$ \begin{align*} \mathrm{spr} (Z ^* ) & = \lim _{k \uparrow \infty} \| \mathrm{Ad} _{Z ^* , Z} ^{(k)} (I_n ) \| ^{\frac{1}{2k}} \nonumber \\[3pt]& \leq \lim \sqrt[\leftroot{-2}\uproot{10}2k]{ \mathrm{tr} \, \mathrm{Ad} _{Z ^*, Z } ^{(k)} (I_n ) } \nonumber \\[3pt]& = \lim \sqrt[\leftroot{-2}\uproot{10}2k]{ \sum _{|\omega | = k} \mathrm{tr} \, Z ^{*\omega} Z^{\omega}} \nonumber \\[3pt]& = \lim \sqrt[\leftroot{-2}\uproot{10}2k]{ \sum _{|\omega | = k} \mathrm{tr} \, Z ^{\omega } Z^{ * \omega }} \nonumber \\[3pt]& = \lim \sqrt[\leftroot{-2}\uproot{10}2k]{ \mathrm{tr} \, \mathrm{Ad} ^{(k)} _{Z , Z^* } (I_n ) } \nonumber \\[3pt]& \leq \lim \sqrt[\leftroot{-2}\uproot{10}2k]{ n \cdot \| Z \| _{\mathrm{row}} ^{2k} } \nonumber \\[3pt]& = \| Z \| _{\mathrm{row}}. \nonumber \end{align*} $$

By [Reference Salomon, Shalit and Shamovich62, Lemma 2.4], the closure of the joint similarity orbit of the row d-tuple, $\mathrm {row} ( Z^* )$ , contains a d-tuple, $W'$ , with norm at most $\| W ' \| _{\mathrm{row}} = \mathrm {spr} (Z^* ) \leq \| Z \| _{\mathrm{row}}$ , and if Z is irreducible, then its joint similarity orbit is closed so that $W'$ is in the joint similarity orbit of $Z^*$ . In particular, given any $\epsilon>0$ , $Z ^*$ is jointly similar to some $W \in \mathbb {C} ^d _n$ with $\| W \| _{\mathrm{row}} \leq \| Z \| _{\mathrm{row}} + \epsilon $ . Viewing $W : \mathbb {C} ^n \otimes \mathbb {C} ^d \rightarrow \mathbb {C} ^n$ as a linear map, its Hilbert space adjoint is $\mathrm {col} (W^*) : \mathbb {C} ^n \rightarrow \mathbb {C} ^n \otimes \mathbb {C} ^d$ with norm

$$ \begin{align*}\| W^* \|_{\mathrm{col}} ^2 = \| W \| _{\mathrm{row}} ^2 \leq (\| Z \| _{\mathrm{row}} + \epsilon ) ^2.\end{align*} $$

Since $\mathrm {row} (Z^*)$ is jointly similar to $\mathrm {row} (W)$ , it follows that $\mathrm {col} (Z)$ is jointly similar to $\mathrm {col} (W ^*)$ where $\| W ^* \| _{\mathrm{col}} \leq \| Z \| _{\mathrm{row}} + \epsilon $ . Proof of the other half of the claim is analogous.

Proof If $\mathfrak {r} \in \mathbb {H} ^2 _d$ , then by [Reference Jury, Martin and Shamovich44, Theorem A], $\mathfrak {r} = K \{ Z , y , v \}$ for some $Z \in \mathbb {B} ^d _n$ and $y,v \in \mathbb {C} ^n$ . Let $\mathfrak {C}$ denote the conjugation (antilinear isometric involution) with respect to the standard basis $\{ e_k \} _{k=1} ^n$ defined by $\mathfrak {C} y = \overline {y}$ , the entrywise (with respect to the standard basis) complex conjugation of y. Given $A \in \mathbb {C} ^d _n$ , let $A^{\mathrm {t}} := (A_1 ^{\mathrm {t}} , \ldots , A_d ^{\mathrm {t}} )$ , where $A_j ^{\mathrm {t}}$ denotes matrix transpose, and let $\overline {A} = \mathfrak {C} A \mathfrak {C}$ denote complex conjugation applied entrywise to A in the standard basis. As described in [Reference Jury, Martin and Shamovich44], since $\mathfrak {r} = K\{ Z , y ,v \}$ for some $Z \in \mathbb {B} ^d _{\mathbb {N}}$ , if we define $A := \overline {Z}$ , $b = \overline {y}$ , and $c = \overline {v}$ , then $(A,b,c)$ is a finite-dimensional realization of $\mathfrak {r}$ . Recall that

$$ \begin{align*} \langle {L^\omega 1} , {K \{ Z , y, v \}} \rangle_{\mathbb{H} ^2} & = \overline{y^* Z^\omega v} \nonumber \\ & = b^* A^\omega c, \nonumber \end{align*} $$

and calculate

$$ \begin{align*} \langle {L^\omega 1} , {U_{\mathrm{t}} K \{ Z , y , v \}} \rangle_{\mathbb{H} ^2} & = \langle {L^{\omega ^{\mathrm{t}}} 1} , { K \{ Z , y , v \}} \rangle_{\mathbb{H} ^2} \nonumber \\& = \left( {b} , {A^{\omega ^{\mathrm{t}}}c} \right) _{\mathbb{C} ^n} \nonumber \\& = \left( {b} , { ((A^{\mathrm{t}}) ^\omega) ^{\mathrm{t}} c} \right) _{\mathbb{C} ^n} \nonumber \\& = \left( {\overline{A} ^{\mathrm{t} \omega} b } , {c} \right) _{\mathbb{C} ^n} \nonumber \\& = \left( {\mathfrak{C} c} , { \mathfrak{C} \overline{A} ^{\mathrm{t} \omega} b} \right) _{\mathbb{C} ^n} \nonumber \\& = \left( {\overline{c}} , {A^{\mathrm{t} \omega} \overline{b}} \right) _{\mathbb{C} ^n}. \nonumber \end{align*} $$

By the previous lemma, the d-tuple $A^{\mathrm {t}} \in \mathbb {C} ^d _n$ is pure, $\mathrm {spr} (A^{\mathrm {t}} ) <1$ , and $A^{\mathrm {t}}$ is jointly similar to a strict column contraction. It follows that $\mathfrak {r} ^{\mathrm {t}}$ has a finite-dimensional realization $(A^{\mathrm {t}} , \overline {c} , \overline {b} )$ , where $A^{\mathrm {t}} = \left ( A_1 ^{\mathrm {t}} , \ldots , A_d ^{\mathrm {t}} \right )$ , and hence $\mathfrak {r} ^{\mathrm {t}} \in \mathbb {H} ^2 _d$ is also NC rational.

Remark 2.3 The domain of any NC rational function, $\mathfrak {r}$ , which is regular at $0$ , is open with respect to the uniform topology on $\mathbb {C} ^d _{\mathbb {N}}$ . Namely, given any $X \in \mathbb {C} ^d _n$ and $Y \in \mathbb {C} ^d _m$ , we define the row pseudometric

$$ \begin{align*} d_{\mathrm{row}} (X,Y ) ^2 & := \| X \otimes I_m - I_n \otimes Y \| ^2 _{\mathrm{row}} = \| (X \otimes I_m - I_n \otimes Y ) ( X \otimes I_m - I_n \otimes Y) ^* \| \nonumber \\ & = \left\| \sum _{k=1} ^d (X_k \otimes I_m - I_n \otimes Y_k ) ( X_k ^* \otimes I_m - I_n \otimes Y_k ^* ) \right\|. \nonumber \end{align*} $$

Since multiplication, summation, and inversion are all jointly continuous in operator norm, and any NC rational function can be constructed by applying finitely many arithmetic operations to free polynomials, it follows that $\mathrm {Dom} \, \mathfrak {r}$ is a uniformly open NC set, and hence contains some row-ball, $t \mathbb {B} ^d _{\mathbb {N}}$ , of nonzero radius $t>0$ . Hence, by rescaling the argument, $\mathfrak {r} _t (Z) := \mathfrak {r} (tZ)$ , we obtain an NC rational function $\mathfrak {r} _r$ with $\overline {\mathbb {B} ^d _{\mathbb {N}}} \subseteq \mathrm {Dom} \, \mathfrak {r} _r$ so that $\mathfrak {r} _t \in \mathbb {H} ^\infty _d$ by [Reference Jury, Martin and Shamovich44, Theorem A]. It follows that given any NC rational function $\mathfrak {r}$ , which is regular at $0$ , there is essentially no loss in generality in assuming that $\mathfrak {r} \in \mathbb {H} ^\infty _d$ , or even $\mathfrak {r} \in [ \mathbb {H} ^\infty _d ] _1$ . Alternatively, if $\mathfrak {r}$ has minimal realization $(A,b,c)$ , then the joint spectral radius of the d-tuple, $A \in \mathbb {C} ^d _n$ , is bounded above by the row norm of A. Hence, $A' := t \cdot A$ , $t ^{-1} := (1 +\epsilon ) \| A \|$ has $\mathrm {spr} (A' ) <1$ , and if $\mathfrak {r} '$ has minimal realization $(A ' , b , c )$ , then $\mathfrak {r} ' (Z) = \mathfrak {r} \left ( t \cdot Z \right )$ is a rescaling of $\mathfrak {r}$ so that $\mathfrak {r} ' \in \mathbb {H} ^\infty _d$ .

2.4 Minimal realizations of $\mathfrak {r} \in \mathbb {H} ^\infty _d$

The minimal realization of any $\mathfrak {r} \in \mathbb {H} ^2 _d$ is easily constructed as follows. Let $c = \mathfrak {r}$ , set

$$ \begin{align*}\mathscr{M} := \bigvee R^{*\omega} \mathfrak{r}, \quad A_k := R^* _k | _{\mathscr{M}},\end{align*} $$

and $b:= P_{\mathscr {M}} 1$ . Since $\mathfrak {r} = K \{ Z ,y , v \}$ is an NC Szegö kernel vector at a finite point $Z \in \mathbb {B} ^d _n$ ,

$$ \begin{align*}\mathscr{M} = \bigvee K \{ Z , y , Z^\omega v \}\end{align*} $$

is finite-dimensional. It is easily checked that the triple $(A,b,c)$ is a realization of $\mathfrak {r}$ . The realization $(A,b,c)$ is controllable by construction, and it is also straightforward to check that it is observable. Alternatively, a minimal realization of $\mathfrak {r}$ can be constructed by applying backward left shifts to $\mathfrak {r} ^{\mathrm {t}}$ .

It will be convenient to also consider Fornasini–Marchesini (FM) realizations of $\mathfrak {r} \in \mathbb {H} ^2 _d$ . Here, an FM realization of $\mathfrak {r} \in \mathbb {H} ^2 _d$ is a quadruple $(A,B,C,D)$ , where $A \in \mathbb {C} ^d _n$ , $B \in \mathbb {C} ^n \otimes \mathbb {C} ^d$ , $C \in \mathbb {C} ^{1\times n}$ , and $D \in \mathbb {C}$ , so that for any $Z \in \mathrm {Dom} \, \mathfrak {r} \supseteq \mathbb {B} ^d _{\mathbb {N}}$ ,

$$ \begin{align*}\mathfrak{r} (Z) = D + C (I-ZA) ^{-1} B.\end{align*} $$

The NC rational function, $\mathfrak {r}$ , is called the transfer function of the FM colligation:

As before, such a realization is called controllable if

$$ \begin{align*}\mathbb{C} ^m = \bigvee _{\substack{\omega \in \mathbb{F} ^d \\ 1 \leq j \leq d }} A^\omega B_j,\end{align*} $$

observable if

$$ \begin{align*}\mathbb{C} ^m = \bigvee A^{*\omega} C ^*,\end{align*} $$

and minimal if N is as small as possible. Again, an FM realization is minimal if and only if it is both observable and controllable, and minimal FM realizations are unique up to joint similarity [Reference Ball, Groenewald and Malakorn7, Theorem 2.1]. It is straightforward to pass back and forth between minimal descriptor and FM realizations. For example, beginning with a minimal descriptor realization $(A,b,c)$ , set $\mathscr {M} _0 := \bigvee _{\omega \neq \emptyset } A^\omega c$ , with projector $P_0$ and $A ^{(0)} := A | _{\mathscr {M} _0}$ . If we define $B_k = A_k c$ , $C := (P_0 b )^*$ , and $D:= \mathfrak {r} (0)$ , then $( A^{(0)}, B , C , D )$ is a minimal FM realization of $\mathfrak {r}$ .

Any $b \in [ \mathbb {H} ^\infty _d ] _1$ has a (generally not finite-dimensional) de Branges–Rovnyak realization [Reference Ball, Bolotnikov and Fang6]. This is a FM-type realization constructed using free de Branges–Rovnyak spaces. Here, given $b \in [ \mathbb {H} ^\infty _d ] _1$ , the right free de Branges–Rovnyak space, $\mathscr {H} ^{\mathrm {t}} (b)$ , is the operator-range space of the operator $\sqrt {I - b(R) b(R) ^*}$ . That is, $\mathscr {H} ^{\mathrm {t}} (b) = \mathrm {Ran} \, \sqrt {I - b(R)b(R) ^*}$ as a vector space, and the norm on $\mathscr {H} ^{\mathrm {t}} (b)$ is defined so that $\sqrt {I - b(R) b(R) ^*}$ is a co-isometry onto its range. Equivalently, $\mathscr {H} ^{\mathrm {t}} (b)$ is the NC-RKHS $\mathcal {H} _{nc} (K^b)$ with CPNC kernel

$$ \begin{align*}K ^b (Z,W) [ \cdot ] := K(Z,W) [ \cdot ] - K(Z,W) [b ^{\mathrm{t}} (Z) ( \cdot ) b^{\mathrm{t}} (W) ^*],\end{align*} $$

and NC kernel vectors

$$ \begin{align*}K^b \{ Z, y ,v \} := (I - b(R) b(R) ^* ) K\{ Z , y ,v \} = K \{ Z , y ,v \} - b(R) K \{ Z ,y , b^{\mathrm{t}} (Z) v \},\end{align*} $$

where $K(Z,W)$ is the CPNC Szegö kernel of the free Hardy space and $K\{Z ,y , v \}$ is an NC Szegö kernel vector. Any right free de Branges–Rovnyak space is contractively contained in $\mathbb {H} ^2 _d$ , and it is always co-invariant for the left free shifts. While $b ^{\mathrm {t}} = b(R) 1$ does not belong to $\mathscr {H} ^{\mathrm {t}} (b)$ in general, $L^* _k b ^{\mathrm {t}}$ always belongs to $\mathscr {H} ^{\mathrm {t}} (b)$ [Reference Ball, Bolotnikov and Fang6, Proposition 4.2]. One then defines the co-isometric de Branges–Rovnyak colligation

where

$$ \begin{align*}\boldsymbol{A} := L^* | _{\mathscr{H} ^{\mathrm{t}} (b)}, \quad \boldsymbol{B} := L^* b^{\mathrm{t}}, \quad \boldsymbol{C} := (K_0 ^b)^*, \quad \mbox{and} \quad \boldsymbol{D} := b(0).\end{align*} $$

One can then check that b is realized as the transfer function of this colligation. In particular, if $b=\mathfrak {b} \in [\mathbb {H} ^\infty _d ] _1$ is an NC rational multiplier, one can cut down this realization to obtain a finite-dimensional and minimal de Branges–Rovnyak FM realization by setting

$$ \begin{align*}\mathscr{M} _0 (\mathfrak{b} ) := \bigvee _{\omega \neq \emptyset} L^{*\omega} \mathfrak{b} ^{\mathrm{t}},\end{align*} $$

with projector $P_0$ and then

where

$$ \begin{align*}\mathfrak{A} := \boldsymbol{A} | _{\mathscr{M} _0 (\mathfrak{b} )}, \quad \mathfrak{B} := \boldsymbol{B}, \quad \mathfrak{C} := \boldsymbol{C} P_0, \quad \mbox{and} \quad \mathfrak{D} := \boldsymbol{D} = \mathfrak{b} (0).\end{align*} $$

Here, note that if $\mathfrak {b} = K\{ Z, y , v \} \in \mathbb {H} ^2 _d$ is NC rational, then $\mathfrak {b} ^{\mathrm {t}}$ is also an NC Szegö kernel at some finite point $W \in \mathbb {B} ^d _n$ by Lemma 2.2 and Theorem A. It immediately follows that $\mathscr {M} _0 (\mathfrak {b} )$ is finite-dimensional.

The proof is routine and omitted. As before, one can alternatively construct a de Branges–Rovnyak realization of any $b \in [ \mathbb {H} ^\infty _d ] _1$ (or minimal de Branges–Rovnyak FM realization of an NC rational $\mathfrak {b} \in [ \mathbb {H} ^\infty _d ] _1$ ) by considering the left free de Branges–Rovnyak space $\mathscr {H} (b)$ , the operator-range space of $\sqrt {I - b(L) b(L) ^*}$ . This space is right shift co-invariant and $R^{*\omega } b \in \mathscr {H} (b)$ for any $\omega \neq \emptyset $ .

2.5 Clark measures

In classical Hardy space theory, there are (essentially) bijections between contractive analytic functions in the disk, Herglotz functions, i.e., analytic functions in $\mathbb {D}$ with positive harmonic real part and positive, finite, and regular Borel measures on the unit circle. Namely, beginning with a positive measure on the circle, $\mu $ , one can define its Herglotz–Riesz integral transform:

$$ \begin{align*}H_\mu (z) := \int _{\partial \mathbb{D}} \frac{1 + z\overline{\zeta}}{1-z\overline{\zeta}} \, \mu (d\zeta ), \quad z \in \mathbb{D},\end{align*} $$

and this produces a Herglotz function in the disk. Note that $\mathrm {Re} \, H_\mu (0) = \mu (\partial \mathbb {D} )> 0$ . Since $\mathrm {Re} \, H_\mu (z) \geq 0$ , applying the so-called Cayley Transform, a fractional linear transformation, which takes the complex right half-plane onto the unit disk, yields a contractive analytic function:

$$ \begin{align*}b_\mu (z) = \frac{H_\mu (z) -1}{H_\mu (z) +1}.\end{align*} $$

Each of these steps is essentially reversible. Beginning with a contractive analytic function $b \in [H^\infty ] _1$ , its inverse Cayley transform,

$$ \begin{align*}H_b (z) := \frac{1+b(z)}{1-b(z)},\end{align*} $$

is a Herglotz function (provided b is not identically equal to $1$ ). Moreover, given any Herglotz function, H, in the disk, the Herglotz representation theorem implies that there is a unique positive measure, $\mu $ , so that

$$ \begin{align*}H(z) = i \mathrm{Im} \, H(0) + H_\mu (z) = i \mathrm{Im} \, H(0) + \int _{\partial \mathbb{D}} \frac{1 + z\overline{\zeta}}{1-z\overline{\zeta}} \, \mu (d\zeta )\end{align*} $$

[Reference Herglotz35]. That is, the Herglotz function corresponding to a positive measure is unique modulo imaginary constants. If $H = H_b$ , this concomitant measure is called the Aleksandrov–Clark measure or Clark measure of b [Reference Aleksandrov1, Reference Aleksandrov2, Reference Clark15]. Hence, any two contractive multipliers $b_1, b_2 \in [ \mathbb {H} ^\infty _d ] _1$ whose Herglotz functions $H_k := H_{b_k}$ differ by an imaginary constant have the same Clark measure. In this case, if $H_2 = H_1 +it$ for some $t \in \mathbb {R}$ , then one can check that

$$ \begin{align*}b_2 = \frac{\overline{z (t)} }{ z(t)} \cdot \mu _{z(t)} \circ b_1\end{align*} $$

is, up to the unimodular constant, $\frac {\overline {z(t)}}{z(t)}$ , a Möbius transformation of $b_1$ corresponding to the point

$$ \begin{align*}z(t) := \frac{t}{2i +t} \in \mathbb{D},\end{align*} $$

so that the contractive analytic functions corresponding to a given positive measure are unique up to such transformations.

By the Riesz–Markov theorem, any positive, finite, and regular Borel measure on $\partial \mathbb {D}$ can be viewed as a positive linear functional on the $C^*$ -algebra of continuous functions on the unit circle, $\mathscr {C} (\partial \mathbb {D} )$ . Recall that the disk algebra, $A (\mathbb {D} )$ , is the unital Banach algebra of analytic functions in the disk which extend continuously to the boundary and that this algebra is isomorphic to the operator algebra $\mathrm {Alg} \{ I , S \} ^{-\| \cdot \|}$ , where $S=M_z : H^2 \rightarrow H^2$ is the shift. By the Weierstrass approximation theorem, $\mathscr {C} (\partial \mathbb {D})$ is the supremum norm-closed linear span of the disk algebra and its conjugates. That is, $\mathscr {C} (\partial \mathbb {D}) = \left ( A(\mathbb {D} ) + A(\mathbb {D} ) ^* \right ) ^{-\| \cdot \|}$ . In the NC multivariable setting of Fock space, the immediate analogue of a positive measure is then any positive linear functional on the norm-closed operator system of the free disk algebra, $\mathbb {A} _d := \mathrm {Alg} \{ I , L_1 , \ldots , L_d \} ^{-\| \cdot \|}$ . We will use the notation

$$ \begin{align*}\mathscr{A} _d := \left( \mathbb{A} _d + \mathbb{A} _d ^* \right) ^{-\| \cdot \|}\end{align*} $$

for the free disk system, and $ \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ will denote the set of positive NC measures, i.e., the set of all positive linear functionals on the free disk system.

As in the single-variable setting, one can define a free Herglotz–Riesz transform of any positive NC measure $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ and this produces an NC Herglotz function, $H_\mu $ , which has positive semidefinite real part in the NC unit row-ball, $\mathbb {B} ^d _{\mathbb {N}}$ . Namely, given $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ and $Z \in \mathbb {B} ^d _n$ , the Herglotz–Riesz transform of $\mu $ is

$$ \begin{align*}H_\mu (Z) := \mathrm{id} _n \otimes \mu \left( (I +ZL^*) (I - ZL^* ) ^{-1} \right), \quad \quad ZL^* := Z_1 \otimes L_1 ^* + \cdots + Z_d \otimes L_d ^*.\end{align*} $$

As before, the Cayley transform of any such $H_\mu $ defines a bijection between NC Herglotz functions and contractive left multipliers, $b_\mu $ , of the Fock space. Furthermore, as before, the correspondence $\mu \leftrightarrow H_\mu $ is bijective modulo imaginary constants, and if a positive NC measure, $\mu $ , corresponds to a contractive left multiplier $b \in [ \mathbb {H} ^\infty _d ] _1$ , we write $\mu = \mu _b$ , and we call $\mu $ the NC Clark measure of b [Reference Jury and Martin38, Reference Jury and Martin39].

3.1 Row isometric dilations of finite row contractions

Let $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ be a finitely correlated Cuntz–Toeplitz functional, and set $\Pi = \Pi _\mu $ . Then $1 + N_\mu $ is $\Pi $ -cyclic so that by Popescu’s NC Wold decomposition, $\Pi = \Pi _L \oplus \Pi _{Cuntz}$ where $\Pi _L$ is unitarily equivalent to L, and $\Pi _{Cuntz}$ is a cyclic and Cuntz (surjective) row isometry [Reference Popescu55, Theorem 1.3]. If we define the finite-dimensional space,

$$ \begin{align*}\mathcal{H} _\mu := \bigvee \Pi ^{*\mu} (1 + N_\mu ),\end{align*} $$

with projection $P_\mu $ , then $\Pi _\mu $ is the minimal row isometric dilation of the finite row contraction

$$ \begin{align*}T_\mu := P_\mu \Pi _\mu | _{\mathcal{H} _\mu \otimes \mathbb{C} ^d}.\end{align*} $$

In particular, $\Pi _\mu $ will be Cuntz if and only if $T_\mu $ is a row co-isometry by [Reference Popescu55, Proposition 2.5]. Let $A := (A_1 , \ldots , A _d ) : \mathcal {H} \otimes \mathbb {C} ^d \rightarrow \mathcal {H}$ be any row contraction on a finite-dimensional Hilbert space $\mathcal {H} \simeq \mathbb {C} ^n$ . Let $V = (V_1 , \ldots , V_d ) : \mathcal {K} \otimes \mathbb {C} ^d \rightarrow \mathcal {K} $ be the minimal row isometric dilation of A on $\mathcal {K} \supsetneqq \mathcal {H}$ . Such row isometries, V, as well as the structure of the unital, WOT-closed algebras they generate, were completely characterized and classified up to unitary equivalence by Davidson, Kribs, and Shpigel in [Reference Davidson, Kribs and Shpigel20]. We will have occasion to apply several results of [Reference Davidson, Kribs and Shpigel20] and so we will record some of the main results of this paper here for future reference.

Given $A, V, \mathcal {H}$ , and $\mathcal {K} $ as above, let $V = V_p \oplus V'$ be the Wold decomposition of V corresponding to $\mathcal {K} = \mathcal {K} _p \oplus \mathcal {K} '$ , where $V_p \simeq L \otimes I_{\mathcal {J}} $ is pure and $V'$ is a Cuntz row isometry on $\mathcal {K} '$ . Furthermore, let $\widetilde {\mathcal {H}}$ be the span of all minimal A-co-invariant subspaces $\widetilde {\mathcal {J} } $ of $\mathcal {H}$ so that $\widetilde {B} := (A ^* | _{\widetilde {\mathcal {J} } } ) ^*$ is a row co-isometry. Then, $\widetilde {\mathcal {H}} = \bigoplus \widetilde {\mathcal {H}} _k$ , where $\{ \widetilde {\mathcal {H}} _k \} _{k=1} ^N$ is a maximal family of mutually orthogonal and minimal $A^*$ -invariant subspaces so that $( A^* | _{\widetilde {\mathcal {H}} _k}) ^*$ is a row co-isometry. Note that if $\widetilde {\mathcal {J} } $ is minimal, then $\widetilde {B}$ is necessarily an irreducible row co-isometry. The following theorem is part of the statement of [Reference Davidson, Kribs and Shpigel20, Theorem 6.5].

In the above statement, recall that a finite row contraction, A, on $\mathcal {H}$ is said to be irreducible, if it has no nontrivial jointly invariant subspace. This is equivalent to $\mathrm {Alg} \{ I, A_1 , \ldots , A_d \} = \mathscr {L} (\mathcal {H} )$ . If A is irreducible, then A cannot have any nontrivial jointly co-invariant subspace either, so that also $\mathrm {Alg} \{ I, A _1 ^* , \ldots , A_d ^* \} = \mathscr {L} (\mathcal {H} )$ . In this case, any vector $x \in \mathcal {H}$ is cyclic for both A and $A^*$ . On the other hand, we say that a row isometry, $V = (V_1, \ldots , V_d )$ , is irreducible if and only if the $V_k$ , $1\leq k \leq d$ , have no nontrivial jointly reducing subspace, i.e., a subspace which is both invariant and co-invariant for each $V_k$ .

The following theorem characterizes when the minimal row isometric dilations of two finite-dimensional row contractions are unitarily equivalent.

We will apply these results to study and characterize the GNS row isometry, $\Pi _\mu $ , arising from a finitely correlated positive NC measure, $\mu $ .

Proof By definition, the finite-dimensional subspace

$$ \begin{align*}\mathcal{H} _\mu = \bigvee \Pi _\mu ^{*\omega} \left(1 + N_\mu\right) = \bigvee T_\mu ^{*\omega} \left(1 + N_\mu\right)\end{align*} $$

is $T_\mu ^*$ -cyclic. Moreover, by the GNS construction of $\mathbb {H} ^2 _d (\mu )$ , $1+N_\mu $ is $\Pi _\mu $ -cyclic. However, since $1 + N_\mu \in \mathcal {H} _\mu $ is $\Pi _\mu $ -cyclic, given any $h \in \mathcal {H} _\mu $ , there is a sequence of polynomials $p_n \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ so that $p_n (\Pi _\mu ) 1 + N _\mu \rightarrow h$ . Hence,

$$ \begin{align*} h & = P _{\mathcal{H} _\mu} h \nonumber \\ & = P _{\mathcal{H} _\mu} \lim p_n (\Pi _\mu ) 1 + N _\mu \nonumber \\ & = \lim p_n (P_{\mathcal{H} _\mu} \Pi _\mu P _{\mathcal{H} _\mu}) 1 + N_\mu \nonumber \\ & = \lim p_n (T_\mu ) 1 + N _\mu, \nonumber \end{align*} $$

since $\mathcal {H} _\mu $ is $\Pi _\mu $ -co-invariant. It follows that $1 +N_\mu $ is also cyclic for $T_\mu $ .

Let $T := (T_1 , \ldots , T_d ) : \mathcal {H} \otimes \mathbb {C} ^d \rightarrow \mathcal {H}$ be any row contraction on a finite-dimensional Hilbert space, $\mathcal {H}$ . Given any $x \in \mathcal {H}$ , define

$$ \begin{align*}\mathcal{H} ' := \bigvee T^{*\omega} x \subseteq \mathcal{H} \quad \mbox{and} \quad T' := \left( T^* | _{\mathcal{H} '} \right) ^*,\end{align*} $$

with projector $P'$ . Finally, define

$$ \begin{align*}\check{\mathcal{H}} := \bigvee T^{ ' \omega} x \subseteq \mathcal{H}',\end{align*} $$

with projector $\check {P}$ and $\check {T} := T' | _{\check {\mathcal {H}} \otimes \mathbb {C} ^d }.$ Observe that $\check {\mathcal {H}}$ is $T'$ -invariant and T-semi-invariant, i.e., it is the direct difference of the nested, T-co-invariant subspaces

$$ \begin{align*}\mathcal{H} ' \quad \mbox{and} \quad \mathcal{H} ' \ominus \check{\mathcal{H}}.\end{align*} $$

Let $\mathcal {K} := \bigvee V^\omega \mathcal {H}$ be the Hilbert space of the minimal row isometric dilation, V, of T; set

$$ \begin{align*}\mathcal{K} _x := \bigvee V^\omega x,\end{align*} $$

with projection, $P_x$ ; and let $V_x := V| _{\mathcal {K} _x}$ .

Proposition 3.6 Given $T,x$ as above, the linear functional, $\mu := \mu _{T,x} \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ , defined by

$$ \begin{align*}\mu _{T,x} (L^\omega ) := \langle {x} , {T^\omega x} \rangle_{\mathcal{H}} = \langle {x} , {\check{T} ^\omega x} \rangle_{\check{\mathcal{H}}},\end{align*} $$

is a finitely correlated positive NC measure. The vector x is both $\check {T}$ and $\check {T} ^*$ -cyclic. The map

$$ \begin{align*}\Pi _\mu ^\omega \left(1 + N_\mu\right) \stackrel{U_x}{\mapsto} V^\omega x\end{align*} $$

is an isometry of $\mathbb {H} ^2 _d (\mu )$ onto $\mathcal {K} _x$ , and $ U_x p(T_\mu ) ^* \left (1 + N_\mu \right ) = P_x p(T) ^*x$ . If x is V-cyclic, then $V \simeq \Pi _\mu $ and $\check {T} \simeq T_\mu $ are unitarily equivalent.

Proof By [Reference Popescu56, Theorem 2.1], the map $L^\omega \mapsto T^\omega $ is completely contractive and unital, and hence extends to a completely positive and unital map of the free disk system into $\mathscr {L} (\mathcal {H} )$ . In particular, $\mu = \mu _{T,x} \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ . The vector x is, by definition, $T^{'*}$ -cyclic, so that

$$ \begin{align*}\bigvee \check{T} ^{*\omega} x = \check{P} \bigvee T^{'*\omega} x = \check{P} \mathcal{H} ' = \check{\mathcal{H}}.\end{align*} $$

This proves that x is $\check {T} ^*$ -cyclic. Since $\check {\mathcal {H}}$ is T-semi-invariant, it follows that $\check {P} T^\omega \check {P} = \check {T} ^\omega $ . Hence,

$$ \begin{align*} \check{\mathcal{H}} & = \bigvee T^{'\omega} x = P' \bigvee T^{\omega} x \nonumber \\ & = \check{P} \bigvee T ^\omega \check{P} x = \bigvee \check{T} ^\omega x, \nonumber \end{align*} $$

so that x is also $\check {T}$ -cyclic. Semi-invariance further implies that $\mu = \mu _{T,x} = \mu _{\check {T} , x}$ .

To see that $\mathfrak {z}^\omega + N_\mu \mapsto V^\omega x$ is an isometry, note that for any free polynomial $p \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ , we can write

$$ \begin{align*}p(L) ^* p(L) = 2 \mathrm{Re} \, q(L) = q(L) + q(L) ^*,\end{align*} $$

for some $q \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ , and then $p(\Pi ) ^* p (\Pi ) = 2 \mathrm {Re} \, q(\Pi )$ for any row isometry $\Pi $ . Hence,

$$ \begin{align*} \| p + N_\mu \| ^2 _\mu & = 2 \mathrm{Re} \, \langle {1+ N_\mu} , {q + N_\mu} \rangle_\mu \nonumber \\ & = 2 \mathrm{Re} \, \langle {x} , {q(V) x} \rangle_{\mathcal{K} } = 2 \mathrm{Re} \, \langle {x} , {q(T) x} \rangle_{\mathcal{H}} \nonumber \\ & = \| p(V) x \| ^2 _{\mathcal{K} } = \| U_x \left(p + N_\mu\right) \| ^2. \nonumber \end{align*} $$

Given any $p(T_\mu ) ^* \left (1 + N_\mu \right ) \in \mathcal {H} _\mu $ , consider

$$ \begin{align*} \langle {\mathfrak{z} ^\omega + N_\mu} , { p(T_\mu ) ^* \left(1 + N_\mu\right)} \rangle_\mu & = \langle { p \mathfrak{z}^{\omega} + N_\mu} , {1+N_\mu} \rangle_\mu \nonumber \\ & = \langle {p(V) V^{\omega} x} , {x} \rangle_{\mathcal{K} } \nonumber \\ & = \langle {V^\omega x} , {P_x p(T) ^* x} \rangle_{\mathcal{K} } \nonumber \\ & = \langle {\mathfrak{z}^\omega + N_\mu} , {U_x ^* P_x p(T) ^* x} \rangle_\mu. \nonumber \end{align*} $$

It follows that

(3.4) $$ \begin{align} U_x p(T_\mu ) ^* \left(1 + N_\mu\right) = P_x p(T) ^* x. \end{align} $$

Finally, if x is V-cyclic, then $\mathcal {K} _x = \mathcal {K} $ and $U_x \Pi _\mu ^\omega = V^\omega U_x$ , so that $U_x$ is an onto isometry, and $\Pi _\mu $ and V are unitarily equivalent. Hence,

$$ \begin{align*}U_x \mathcal{H} _\mu = U_x \bigvee \Pi _\mu ^{*\omega } \left(1+N_\mu\right) = \bigvee V^{*\omega} x = \bigvee T^{*\omega} x = \mathcal{H} '.\end{align*} $$

Moreover, since x is V-cyclic, given any $h \in \mathcal {H}'$ , there is a sequence of polynomials $p_n \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ so that $p_n (V) x \rightarrow h$ , and then

$$ \begin{align*}h = \lim p_n (V) x = P ' h = \lim P' p_n (V) x = \lim p_n (T' ) x,\end{align*} $$

so that x is also $T'$ -cyclic, $\check {\mathcal {H}} = \mathcal {H} '$ , and $\check {T} = T'$ . It further follows that $T' = \check {T}$ and $T_\mu $ are unitarily equivalent via $U_x$ . Indeed, if $\mathcal {K} _x = \mathcal {K} $ so that $P_x = I$ , then equation (3.4) becomes

$$ \begin{align*}U_x p(T _\mu ) ^* ( 1 + N_\mu ) = p(T) ^* x.\end{align*} $$

In particular, since $U_x$ restricts to a unitary map from $\mathcal {H} _\mu $ onto $\mathcal {H} '$ , we obtain that for any $y _\mu := q (T_\mu ) ^* (1 + N _\mu ) \in \mathcal {H} _\mu $ , $q \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ and $1\leq k \leq d$ ,

$$ \begin{align*}U_x T_{\mu; k} ^* y _\mu = T_k ^* q(T) ^* x = T_k ^* U_x y_\mu.\end{align*} $$

This proves that $T_\mu $ is unitarily equivalent to $T '$ .

By [Reference Popescu55, Proposition 2.5], a row contraction is a row co-isometry if and only if its minimal row isometric dilation is a Cuntz row isometry. If $V := \left ( V_1 , \ldots , V _d \right )$ is a row isometry on a separable Hilbert space, $\mathcal {K} $ , recall that $\mathfrak {V} := \mathrm {Alg} \{ I , V _1 , \ldots , V_d \} ^{-WOT}$ is called the free semigroup algebra of V [Reference Davidson19].

Proof If T is an irreducible row co-isometry, then $\mathcal {H}$ is the unique, minimal T-co-invariant subspace of $\mathcal {H}$ so that V is irreducible by [Reference Davidson, Kribs and Shpigel20, Lemma 5.8]. Moreover, by [Reference Davidson, Kribs and Shpigel20, Theorem 5.2], since T is irreducible, the $WOT$ -closed unital algebra of V contains $P _{\mathcal {H}}$ , the projection onto $\mathcal {H}$ . Again, since T is irreducible, given any fixed nonzero $x \in \mathcal {H}$ , any $h \in \mathcal {H}$ can be written as $h=p(T) x$ for some $p \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ , so that

$$ \begin{align*} h & = p(T) x \nonumber \\ & = \underbrace{P _{\mathcal{H}} p(V)}_{\in \mathfrak{V}} x. \nonumber \end{align*} $$

Hence, any $h \in \mathcal {H}$ belongs to the weak and hence Hilbert space norm closure of $\mathfrak {V} x$ , so that

$$ \begin{align*}\bigvee \mathfrak{V} x = \bigvee \mathfrak{V} \mathcal{H} = \mathcal{K} ,\end{align*} $$

since V is the minimal row isometric dilation of T.

Proof If h is V-cyclic, then it is clearly also T-cyclic. Conversely, if h is T-cyclic, consider the space

$$ \begin{align*}\mathcal{K} (h) := \bigvee V^\omega h \subseteq \mathcal{K} .\end{align*} $$

If h is not V-cyclic, then $\mathcal {K} (h) \subsetneqq \mathcal {K} $ , and there is a nonzero $x \in \mathcal {K} (h) ^\perp $ and $\mathcal {K} (h) ^\perp $ is V-co-invariant. By [Reference Davidson, Kribs and Shpigel20, Corollary 4.2], there is a nonzero $g \in \mathcal {H} \bigcap \bigvee V^{*\omega } x \subseteq \mathcal {K} (h) ^\perp $ . Hence, $V^{*\omega } g = T^{*\omega } g \perp \mathcal {K} (h)$ for any $\omega \in \mathbb {F} ^d$ . However, by assumption, h is T-cyclic so that $g=p(T) h$ for some $p \in \mathbb {C} \{ \mathbb {\mathfrak {z}} \} $ and

$$ \begin{align*}\| g\| ^2 = \langle {p(T) h} , {g} \rangle = \langle {h} , {p(T) ^* g} \rangle =0,\end{align*} $$

contradicting that $g\neq 0$ .

In [Reference Kennedy50], Kennedy refined the Wold decomposition of any row isometry by further decomposing any Cuntz row isometry into the direct sum of three types: Cuntz type-L (or absolutely continuous Cuntz [ACC]), von Neumann type, and dilation type.

Remark 3.11 Any dilation-type row isometry, $\Pi $ , has an upper triangular decomposition of the form

so that $\Pi $ has a restriction to an invariant subspace which is unitarily equivalent to a pure row isometry and $\Pi $ is the minimal row isometric dilation of its compression, T, to the orthogonal complement of this invariant space [Reference Kennedy50, Proposition 6.2]. Since $\Pi $ is of Cuntz type, T is necessarily a row co-isometry [Reference Popescu55, Proposition 2.5].

Recall that a vector, $h \in \mathcal {H}$ , is said to be a wandering vector for a row isometry $V : \mathcal {H} \otimes \mathbb {C} ^d \rightarrow \mathcal {H}$ , if

$$ \begin{align*}\langle {V^\alpha h} , {V^\omega h} \rangle_{\mathcal{H}} = \delta _{\alpha , \omega} \| h \| ^2 _{\mathcal{H}},\end{align*} $$

and that the closed linear span of all wandering vectors for V is $\mathrm {Ran} \, V ^\perp $ . If x is a unit wandering vector for V, then

$$ \begin{align*}\mathcal{H} _x := \bigvee _{\omega \in \mathbb{F} ^d} V^\omega x\end{align*} $$

is V-invariant and the linear map, $U_x : \mathcal {H} _x \rightarrow \mathbb {H} ^2 _d$ , defined by $U_x V^\omega x := L^\omega 1$ is an onto isometry intertwining V and L, $U_x V^\omega = L^\omega U_x$ [Reference Popescu55].

Proof By [Reference Davidson, Kribs and Shpigel20, Corollary 4.2], $\mathcal {H} ' := \mathcal {K} ' \bigcap \mathcal {H} \neq \{ 0 \}$ . Define the subspace

$$ \begin{align*}\mathscr{W} ' := \left( \mathcal{H}' + \bigvee _{j=1} ^d V_j \mathcal{H}' \right) \ominus \mathcal{H} '.\end{align*} $$

By [Reference Davidson, Kribs and Shpigel20, Lemma 3.1], this is a nontrivial wandering subspace for V.

Proof Lemma 3.12 implies that any direct summand of V has wandering vectors. It is then an immediate consequence of [Reference Kennedy49, Corollary 4.13] that V has no direct summand of von Neumann type.

Suppose that V had a nontrivial ACC direct summand, $V_{ac}$ , acting on the V-reducing subspace $\mathcal {K} _{ac}$ . Then, by [Reference Davidson, Kribs and Shpigel20, Corollary 4.2], $\mathcal {H} _{ac} := \mathcal {K} _{ac} \bigcap \mathcal {H} \neq \{ 0 \}$ , and $\mathcal {H} _{ac}$ is $V_{ac}$ -co-invariant. Let $T_{ac} := \left ( V_{ac} ^* | _{\mathcal {H} _{ac} } \right ) ^*$ , then $V_{ac}$ is the minimal row isometric dilation of $T_{ac}$ . Indeed,

$$ \begin{align*}\widetilde{\mathcal{K} } _{ac} := \bigvee _\omega V_{ac} ^\omega \mathcal{H} _{ac}\end{align*} $$

is $V_{ac}$ -reducing, so that $\mathcal {K} _{ac} \ominus \widetilde {\mathcal {K} } _{ac} =: \mathcal {K} ' _{ac}$ is also $V_{ac}$ -reducing. Then, by [Reference Davidson, Kribs and Shpigel20, Corollary 4.2], $\mathcal {K} ' _{ac} \bigcap \mathcal {H} = \mathcal {H} ' _{ac} \subseteq \mathcal {H} _{ac}$ is nontrivial. This contradicts that $\mathcal {K} ' _{ac} \perp \widetilde {\mathcal {K} } _{ac}$ , and we conclude that $\widetilde {\mathcal {K} } _{ac} = \mathcal {K} _{ac}$ .

Let $\{ \mathcal {H} _{ac; k} \}$ be a maximal family of minimal, pairwise orthogonal, and $V_{ac}$ -co-invariant subspaces of $\mathcal {H} _{ac}$ . Fix some k and choose any nonzero $h \in \mathcal {H} _{ac; k}$ . Then, since $V_{ac}$ is absolutely continuous, every $y \in \mathcal {K} _{ac}$ is an absolutely continuous vector for $V_{ac}$ in the sense of [Reference Davidson, Li and Pitts21, Definition 2.4]. In particular, by [Reference Davidson, Li and Pitts21, Theorem 2.7], $h \in \mathcal {H} _{ac ;k} \subsetneq \mathcal {K} _{ac}$ is in the range of a bounded intertwiner, $X: \mathbb {H} ^2 _d \rightarrow \mathcal {K} _{ac}$ . That is, $X L_k = V_{ac; k} X$ , and there is a $g \in \mathbb {H} ^2 _d$ so that $Xg =h$ . Note that $X^*$ must be bounded below on $\mathcal {H} _{ac; k}$ . First, $\mathrm {Ker} \, X^* $ is $V_{ac}$ -co-invariant since $X^* x =0$ implies that

$$ \begin{align*}X^* V_{ac} ^{*\omega} x = L^{*\omega} X^* x =0.\end{align*} $$

The subspace $\mathcal {H} _{ac; k}$ is finite-dimensional, so that if $X^*$ is not bounded below on this space, then it has nontrivial kernel. However, if

$$ \begin{align*}\ker X^* \bigcap \mathcal{H} _{ac; k } \neq \{ 0 \},\end{align*} $$

then this is a proper, nontrivial $V_{ac}$ -co-invariant subspace of $\mathcal {H} _{ac; k}$ , contradicting the minimality of $\mathcal {H} _{ac; k}$ . Hence, $X^*$ is bounded below by say $\epsilon>0$ on $\mathcal {H} _{ac; k}$ . Also recall that since $V_{ac}$ is Cuntz, $T_{ac}$ is a row co-isometry. Then, for any $n \in \mathbb {N}$ ,

$$ \begin{align*} \| h \| ^2 & = \sum _{|\omega| = n} \| T_{ac} ^{*\omega} h \| ^2 \nonumber \\ & \leq \epsilon ^{-2} \sum _{|\omega| = n} \| X^* T_{ac} ^{*\omega} h \| ^2 \nonumber \\ & = \epsilon ^{-2} \sum _{|\omega| = n} \| X^* V_{ac} ^{*\omega} h \| ^2 \nonumber \\ & = \epsilon ^{-2} \sum _{|\omega| = n} \| L^{*\omega} X^* h \| ^2 \nonumber \\ &\rightarrow 0, \nonumber \end{align*} $$

since L is pure. This contradiction proves the claim.

Alternatively, if $V_{ac}$ is an ACC direct summand of V acting on $\mathcal {K} _{ac}$ , then $\mathcal {H} _{ac} := \mathcal {H} \cap \mathcal {K} _{ac}$ is nontrivial and $V_{ac}$ -co-invariant by [Reference Davidson, Kribs and Shpigel20, Corollary 4.2]. If $\mathcal {H} _{ac} ' \subseteq \mathcal {H} _{ac}$ is any minimal $V_{ac}$ -co-invariant subspace, then $\mathcal {K} ' _{ac} := \bigvee V_{ac} ^\omega \mathcal {H} ' _{ac}$ is $V_{ac}$ -reducing and we set $V_{ac} ' := V_{ac} | _{\mathcal {K} ' _{ac}}$ . By [Reference Davidson, Kribs and Shpigel20, Lemma 5.4], since $\mathcal {H} ' _{ac}$ is a minimal $V_{ac}'$ -co-invariant subspace that is cyclic for $V_{ac} '$ , the free semigroup algebra, $\mathfrak {V} ' _{ac} = \mathrm {Alg} \{ I , V_{ac;1} ' , \ldots , V_{ac ;d } ' \} ^{-WOT}$ , contains the projection, $P_{ac} '$ onto $\mathcal {H} _{ac} '$ . However, $V_{ac}$ and hence $V_{ac} '$ are absolutely continuous, so that $\mathfrak {V} ' _{ac}$ is completely isometrically isomorphic and weak- $*$ homeomorphic to $\mathscr {L} ^\infty _d$ , the left analytic Toeplitz algebra. This produces a contradiction as $\mathscr {L} ^\infty _d$ contains no nontrivial projections by [Reference Davidson and Pitts23, Corollary 1.5].

Any row isometry, V, has the Kennedy–Lebesgue–von Neumann–Wold decomposition $V = V_L \oplus V_{ACC} \oplus V_{dil} \oplus V_{vN}$ , and we have shown that if V is the minimal row isometric dilation of a finite row contraction, then the ACC and von Neumann-type direct summands are absent.

The analogue of normalized Lebesgue measure in this setting is NC Lebesgue measure, $m (L^\omega ) := \langle {1} , {L^\omega 1} \rangle _{\mathbb {H} ^2}$ , the so-called vacuum state of the Fock space. (This NC measure is the NC Clark measure of the identically $0$ multiplier, just as normalized Lebesgue measure on the circle is the Clark measure of the identically $0$ function in the disk.) In [Reference Jury and Martin40, Reference Jury and Martin41], the first two authors have constructed the Lebesgue decomposition of any positive NC measure $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ with respect to NC Lebesgue measure, m. In particular, $\mu $ is singular with respect to NC Lebesgue measure in the sense of [Reference Jury and Martin40, Reference Jury and Martin41] if and only if its GNS row isometry is the direct sum of dilation-type and von Neumann-type row isometries [Reference Jury and Martin41, Corollary 8.13]. We say a positive NC measure $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ is of a given type if its GNS row isometry is of that corresponding type. The GNS space of $\mu $ decomposes as the direct sum,

$$ \begin{align*}\mathbb{H} ^2 _d (\mu ) = \mathbb{H} ^2 _d (\mu _{ac} ) \oplus \mathbb{H} ^2 _d (\mu _s ),\end{align*} $$

and $\Pi _{\mu } = \Pi _{\mu _{ac}} \oplus \Pi _{\mu _s }$ with respect to this direct sum. Here,

$$ \begin{align*}\Pi _{\mu _{ac}} = \Pi _{\mu; L} \oplus \Pi _{\mu; ACC} \quad \mbox{and} \quad \Pi _{\mu _s} = \Pi _{\mu; dil} \oplus \Pi _{\mu; vN}\end{align*} $$

(see [Reference Jury and Martin41, Section 8]).

A bounded operator, $T \in \mathscr {L} (\mathbb {H} ^2 _d )$ , is called left Toeplitz if $L_j ^* T L_k = \delta _{j,k} I$ . Such operators are called multi-Toeplitz in [Reference Popescu57]. Here, recall that a bounded operator, T, on the Hardy space, $H^2 (\mathbb {D} )$ , is called Toeplitz if $T = T_f = P_{H^2} M_f | _{H^2}$ for some $f \in \mathscr {L} ^\infty (\partial \mathbb {D} )$ . A result of Brown and Halmos identifies the bounded Toeplitz operators as the set of all bounded operators $T \in \mathscr {L} (H^2 )$ with the Toeplitz property:

$$ \begin{align*}S^* T S = T,\end{align*} $$

where $S = M_z$ is the shift on $H^2$ [Reference Halmos and Brown28, Theorem 6]. If $b \in [H^\infty ] _1$ , then $T := I - b(S) ^* b(S) \geq 0$ is a positive semidefinite Toeplitz operator, and b is not an extreme point of the closed convex set $[ H^\infty ] _1$ if and only if there is a unique, outer $a \in [ H^\infty ]_1$ , the Sarason function of b, so that $a(0)>0$ and the column $c := \left ( \begin {smallmatrix} b \\ a \end {smallmatrix} \right ) $ is inner. A contractive left multiplier of Fock space, $b \in [\mathbb {H} ^\infty _d ]_1$ , is said to be column-extreme (CE), if contractivity of the column left multiplier, $c := \left ( \begin {smallmatrix} b \\ a \end {smallmatrix} \right ) $ , for $a \in \mathbb {H} ^\infty _d$ implies $a \equiv 0$ [Reference Jury and Martin39]. In [Reference Jury and Martin39], we observed that any CE b is necessarily an extreme point, and that if b is non-CE, then one can define a unique Sarason outer function, $a \in [\mathbb {H} ^\infty _d ] _1$ , so that $a(0)>0$ and $c := \left ( \begin {smallmatrix} b \\ a \end {smallmatrix} \right ) $ is CE. In [Reference Jury and Martin42], we proved that a is outer, and that if $b = \mathfrak {b}$ is NC rational and non-CE, then $a = \mathfrak {a}$ is NC rational and the column $\mathfrak {c} := \left ( \begin {smallmatrix} \mathfrak {b} \\ \mathfrak {a} \end {smallmatrix} \right ) $ is inner.

Theorem 3.14 Let $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ be a finitely correlated Cuntz–Toeplitz functional with NC Lebesgue decomposition $\mu = \mu _{ac} + \mu _s$ . The absolutely continuous part of $\mu $ , $\mu _{ac} = \mu _L$ , is purely of type-L and $\Pi _{\mu _L} \simeq L$ . If $\mathfrak {b} \in [\mathbb {H} ^\infty _d ] _1$ is the contractive NC rational left multiplier so that $\mu = \mu _{\mathfrak {b}}$ is the NC Clark measure of $\mathfrak {b}$ , then

$$ \begin{align*}\mu _{ac} (L^\omega ) = \langle {1} , {(I -\mathfrak{b} (R) ^* ) ^{-1} \mathfrak{a} (R) ^* \mathfrak{a} (R) (I -\mathfrak{b} (R) ) ^{-1} L^\omega 1} \rangle_{\mathbb{H} ^2},\end{align*} $$

where $\mathfrak {a} \in [ \mathbb {H} ^\infty _d ] _1$ is the contractive outer NC rational Sarason function of $\mathfrak {b}$ ,

$$ \begin{align*}T := (I -\mathfrak{b} (R) ^* ) ^{-1} \mathfrak{a} (R) ^* \mathfrak{a} (R) (I -\mathfrak{b} (R) ) ^{-1}\end{align*} $$

is a bounded left Toeplitz operator, and $\mathfrak {a} (1 - \mathfrak {b} ) ^{-1} \in \mathbb {H} ^\infty _d$ .

The singular part, $\mu _{s}$ , is purely of dilation-type and $\Pi _{\mu; s} = \Pi _{\mu; dil} = \bigoplus _{j=1} ^N \Pi ^{(j)}$ acting on $\mathbb {H} ^2 _d (\mu _{dil} ) = \bigoplus _{j=1} ^N \mathcal {K} _j$ is the direct sum of at most finitely many irreducible Cuntz row isometries of dilation-type. If $T^{(j) *} := \Pi ^{(j)*} | _{\mathcal {H} _\mu \cap \mathcal {K} _j}$ , then each $T^{(j)}$ is a finite and irreducible row co-isometry with irreducible and minimal row isometric dilation $\Pi ^{(j)}$ .

Of course, it may be that either $\mu _{ac} =0$ or $\mu _s =0$ .

Remark 3.17 If $\mu \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ is any positive NC measure, we can define a positive extension of $\mu $ from the free disk system to the Cuntz–Toeplitz $C^*$ -algebra, $\mathcal {E} _d = C^* \{ I , L_1 , \ldots , L_d \}$ , by

$$ \begin{align*}\hat{\mu} ( a_1 ( L ) a_2 (L) ^* ) := \langle {1 + N_\mu} , {\pi _\mu (a_1) \pi _\mu (a_2 ) ^* \left(1 + N_\mu\right)} \rangle_\mu,\end{align*} $$

where $\pi _\mu : \mathcal {E} _d \rightarrow \mathscr {L} (\mathbb {H} ^2 _d (\mu ) )$ is the GNS $*$ -representation obtained from $\mu $ . By well-known results in $C^*$ -algebra theory, assuming that $\mu $ is unital, i.e., a state, $\hat {\mu }$ will be an extreme point in the state space of $\mathcal {E} _d$ , i.e., a pure state, if and only if $\pi _\mu $ is irreducible [Reference Davidson18, Theorem I.9.8]. Equivalently, $H_\mu $ will be an extreme point in the set of all NC Herglotz functions obeying $H(0) =1$ .

Remark 3.18 Let $\mathfrak {b}$ be NC rational inner. That $\mu _{\mathfrak {b}} \in \left ( \mathscr {A} _d \right ) ^{\dagger } _+ $ is a finitely correlated Cuntz functional is an analogue of classical theory. Indeed, any rational inner $\mathfrak {b} \in H^2$ is a finite Blaschke product,

$$ \begin{align*}\mathfrak{b} (z) = \zeta \, \prod _{k=1} ^N \frac{z-w_k}{1-\overline{w}_k z}, \quad \quad w_k \in \mathbb{D}, \ \zeta \in \partial \mathbb{D}.\end{align*} $$

In this case, the Clark measure, $\mu _{\mathfrak {b}}$ , is a finite positive linear combination of exactly N Dirac point masses. This singular measure, $\mu _{\mathfrak {b}}$ , is supported on the set of points, $\zeta \in \partial \mathbb {D}$ , at which $\mathfrak {b} (\zeta ) =1$ , so that the point masses are located at the N roots of the degree N polynomial,

$$ \begin{align*}\prod _{k=1} ^N (z-w_k) - \prod _{k=1} ^N (1 - \overline{w} _k z).\end{align*} $$

A singular finitely correlated functional can then be thought of as an analogue of a positive linear combination of finitely many point masses. If the GNS representation of the functional is irreducible, this can be interpreted as the analogue of a single atom. In this case, where $d=1$ , if $\mu $ is such a finite linear combination of point masses, then $H^2 (\mu ) = L^2 (\mu )$ so that $\Pi _\mu := M_\zeta | _{H^2 (\mu )}$ is unitary, and

$$ \begin{align*}\mathcal{H} _\mu := \bigvee _{k\geq 0} M_\zeta ^{*k} 1 = H^2 (\mu ).\end{align*} $$

In this case, $\mathcal {H} _\mu = H^2 (\mu )$ is finite-dimensional (of dimension $=N$ if $\mu $ is a linear combination of N point masses) and $\Pi _\mu $ is a finite-dimensional unitary. Indeed, since $T_\mu $ is then a finite co-isometry, it must be unitary, so that $T _\mu = \Pi _\mu $ in this single-variable case. In this regard, the theory becomes more complicated when $d>1$ as there are no finite-dimensional row isometries.

Finally, suppose that a contractive rational multiplier $\mathfrak {b} \in [ H^\infty ] _1$ is not an extreme point. Then $1 - |\mathfrak {b} (\zeta ) | ^2 = |\mathfrak {a} (\zeta ) | ^2$ , $ a.e. \ \partial \mathbb {D}$ , where $\mathfrak {a}$ is the rational Sarason function of $\mathfrak {b}$ , and it follows that $M_\zeta | _{H^2 (\mu _{\mathfrak {b}; ac})}$ is a pure cyclic isometry unitarily equivalent to the shift. In particular, $H^2 ( \mu _{\mathfrak {b} } )$ is infinite-dimensional. Recall that there is onto isometry, the weighted Cauchy transform, $\mathscr {F} _{\mathfrak {b}} : H^2 (\mu _{\mathfrak {b}} ) \rightarrow \mathscr {H} (\mathfrak {b} )$ of $H^2 (\mu _b )$ onto the de Branges–Rovnyak space of $\mathfrak {b}$ , and that the image of $\Pi _{\mathfrak {b}} ^*$ , where $\Pi _{\mathfrak {b}} := M_\zeta | _{H^2 (\mu _{\mathfrak {b}} )}$ , under the weighted Cauchy transform is a rank-one perturbation of the restricted backward shift,

$$ \begin{align*}X(1) := \underbrace{S^* | _{\mathscr{H} (\mathfrak{b} )}}_{=: X} + \frac{1}{1- \mathfrak{b} (0) } \langle {K_0 ^{\mathfrak{b}}} , {\cdot} \rangle_{\mathscr{H} (\mathfrak{b} )} S^* \mathfrak{b},\end{align*} $$

$X(1) = \mathscr {F} _{\mathfrak {b}} \Pi _{\mathfrak {b}} ^* \mathscr {F} _{\mathfrak {b}} ^*$ [Reference Clark15]. In this case, since $\mathfrak {b}$ is rational, we obtain that

$$ \begin{align*}\mathcal{H} _\mu := \bigvee \Pi _{\mathfrak{b}} ^{*k} 1 \simeq \bigvee X(1) ^{k} K_0 ^{\mathfrak{b}} =: \mathscr{M} (\mathfrak{b} ) \subsetneqq \mathscr{H} (\mathfrak{b} )\end{align*} $$

is a finite-dimensional subspace of $\mathscr {H} (\mathfrak {b} )$ . Hence, in this case, if P is the projection onto $\mathscr {M} (\mathfrak {b} )$ , then

$$ \begin{align*}T (1) := \left( X(1) | _{\mathscr{M} (\mathfrak{b} ) } \right) ^*\end{align*} $$

is a finite-dimensional contraction, and $X(1) ^* \simeq \Pi _{\mathfrak {b}} = M _\zeta | _{H^2 (\mu _{\mathfrak {b}} )}$ is its minimal isometric dilation. This again shows that our results are natural extensions of classical theory.

5.1 Boundary values

Let $\mathfrak {b} \in [ \mathbb {H} ^\infty _d ] _1$ be NC rational. We further assume, without loss of generality, that $\mathfrak {b} (0) =0$ .

In the above statement, $T (\zeta ) ^{\mathrm {t}} = \left ( T (\zeta ) _{ 1} ^{\mathrm {t}}, \ldots , T (\zeta ) _{d } ^{\mathrm {t}} \right )$ , where $\mathrm {t}$ denotes matrix transpose of each component of $T(\zeta )$ with respect to a choice of orthonormal basis of $\mathscr {M} (\mathfrak {b} )$ .