Słociński–Wold decompositions for row isometries

Abstract

Słociński gave sufficient conditions for commuting isometries to have a nice Wold-like decomposition. In this note, we provide analogous results for row isometries satisfying certain commutation relations. Other than known results for doubly commuting row isometries, we provide sufficient conditions for a Wold decomposition based on the Lebesgue decomposition of the row isometries.

1 Introduction

Let V be an isometry acting on a Hilbert space H. A well-known result, discovered independently by von Neumann (1929) and Wold (1938), tells us that H decomposes uniquely into V-reducing subspaces $H=H_u\oplus H_s$ where $V|_{H_u}$ is a unitary and $V_{H_s}$ is a unilateral shift. We will follow the convention of calling this result the Wold decomposition of V. Over the decades, there have been generalizations of this result, decomposing isometric representations of semigroups into their unitary and nonunitary parts. Suciu’s work in [20] is an early example of such results.

The work at hand is largely inspired by the Wold-like decomposition given Słociński [19]. Let $V_1$ and $V_2$ be commuting isometries on a Hilbert space H. We say that $V_1$ and $V_2$ have a Słociński–Wold decomposition if H decomposes as $H = H_1 \oplus H_2 \oplus H_3 \oplus H_4$ , where each space $H_i$ reduces both $V_1$ and $V_2$ ; $V_1|_{H_1},V_1|_{H_2},V_2|_{H_1},V_2|_{H_3}$ are unitaries; and $V_1|_{H_3},V_1|_{H_4},V_2|_{H_2},V_2|_{H_4}$ are unilateral shifts. Słociński gives sufficient conditions for a pair commuting isometries to have a Słociński–Wold decomposition. Most notable, or at least the most noted, of these results is that a pair of doubly commuting isometries $V_1$ and $V_2$ has a Słociński–Wold decomposition (where doubly commuting means that $V_1V_2=V_2V_1$ and $V_1^*V_2 = V_2V_1^*$ ). Generalizations of this result for n doubly commuting isometries have been given [8]. Słociński also gives sufficient conditions for the existence of a Słociński–Wold decomposition based on the structure of the individual unitary parts of the isometries. Recall that a unitary U can decomposed as $U_{\mathrm{abs}} \oplus U_{\mathrm{sing}}$ where $U_{\mathrm{abs}}$ has absolutely continuous spectral measure and $U_{\mathrm{sing}}$ has singular spectral measure (both with respect to Lebesgue measure). Słociński gives two results [19, Theorems 4 and 5], showing the existence of a Słociński–Wold decomposition in the absence of absolutely continuous unitary parts.

Let $S = [S_1,\ldots ,S_m]$ be a row isometry on a Hilbert space H. That is, $S \colon H^{(m)} \rightarrow H$ is an isometric map. Equivalently, $S = [S_1,\ldots ,S_m]$ is a row isometry if $S_1, \ldots , S_n$ are isometries with pairwise orthogonal ranges. Popescu [14] shows that there is a Wold decomposition for S. That is, H can be decomposed into S-reducing subspaces $H = H_u \oplus H_s$ where $S|_{H_u}$ is a row unitary, and $S|_{H_s}$ is an n-shift. Beyond row isometries, Muhly and Solel [13] give a Wold decomposition for isometric representations of C $^*$ -correspondences, decomposing an isometric representation into unitary and induced parts.

Let $S=[S_1,\ldots ,S_m]$ and $T=[T_1,\ldots , T_n]$ be two row isometries on a Hilbert space H. We say that S and $T \ \theta $ -commute if there is a permutation $\theta \in S_{m\times n}$ such that for $1\leq i \leq m$ and $1\leq i \leq n$ , $S_iT_j = T_{j'}S_{i'}$ when $\theta (i,j) = (i',j')$ . A pair of $\theta $ -commuting row isometries determines an isometric representation of a $2$ -graph with a single vertex. Thus, a pair of $\theta $ -commuting row isometries is an isometric representation of a product system of two finite-dimensional C $^*$ -correspondences (see, e.g., [6, Section 4]). Skalski and Zacharias [18] generalized Słociński’s Wold decomposition for doubly commuting isometries to isometric representations of product systems of C $^*$ -correspondences which satisfy a doubly commuting condition. Thus, Skalski and Zacharias’s result gives a Słociński–Wold decomposition for $\theta $ -commuting row isometries.

In this note, we will give sufficient conditions for two $\theta $ -commuting row isometries to have a Słociński–Wold decomposition mirroring the three theorems proved by Słociński for commuting isometries. Theorems 3–5 of [19] are generalized in Theorems 3.4, 3.8, and 3.10, respectively. In [19, Theorems 4 and 5], Słociński uses the Lebesgue decomposition of a unitary. For row unitaries, we use the Lebesgue decomposition due to Kennedy [10]. This states that any row unitary decomposes into an absolutely continuous row unitary, a singular row unitary, and a third part called a dilation-type row unitary. For a single unitary U, the statements “U has no absolutely continuous part” and “U is singular” are equivalent; for row unitaries, the existence of dilation-type parts means that the latter is a stronger statement than the former. In this note, for a row unitary, the statement “U is singular” will play the role that “U has no absolutely continuous part” played in [19].

2 Row isometries and their structure

A row isometry on a Hilbert space H is an isometric map S from $H^{(n)}$ to H. An operator $S \colon H^{(n)} \rightarrow S$ is a row isometry if and only if $S = [S_1, \ldots , S_m]$ where $S_1, \ldots , S_m$ are isometries on H with pairwise orthogonal ranges. Equivalently, the $S_1, \ldots , S_m$ are isometries satisfying

$$ \begin{align*} \sum_{i=1}^m S_iS_i^* \leq I_H. \end{align*} $$

A row isometry $S = [S_1, \ldots , S_m]$ is a row unitary if S is a unitary map. Equivalently, S is a row unitary if

$$ \begin{align*} \sum_{i=1}^m S_iS_i^* = I_H. \end{align*} $$

Let $S=[S_1, \ldots , S_m]$ be a row operator on a Hilbert space H, and let $M \subseteq H$ be a subspace. The subspace M is S-invariant if $S_i H \subseteq H$ for each $1\leq i \leq m$ ; M is $S^*$ -invariant if $S_i^* H \subseteq H$ for each $1\leq i \leq m$ ; and M is S-reducing if M is both S-invariant and $S^*$ -invariant.

Denote by $\mathbb {F}_m^+$ the unital free semigroup on n generators $\{1,\ldots , m\}$ . For $w = w_1 \ldots w_k \in \mathbb {F}_n^+$ , denote by $S_w$ the isometry

$$ \begin{align*} S_{w_1}S_{w_2}\ldots S_{w_k}. \end{align*} $$

Here, $S_\emptyset $ will denote $I_H$ .

Example 2.1 Let $H= \ell ^2(\mathbb {F}_m^+)$ with orthonormal basis $\{ \xi _w \colon w \in \mathbb {F}_m^+ \}.$ For $i \in \{1, \ldots , m\}$ , define the operator $L_i$ by

$$ \begin{align*} L_i \xi_w = \xi_{iw}. \end{align*} $$

Then $L = [L_1, \ldots , L_m]$ is a row isometry on H.

Definition 2.1 Let $S = [S_1, \ldots , S_m]$ be a row isometry. Let L be the row isometry described in Example 2.1. We call S an m-shift of multiplicity $\alpha $ if S is unitarily equivalent to an ampliation of L by $\alpha $ . That is, $[S_1,\ldots ,S_m] \simeq [L_1^{(\alpha )},\ldots ,L_m^{(\alpha )}]$ .

Note that when $m=1$ , an m-shift is a unilateral shift. Thus, the following result, due to Popescu [14], is a generalization of the Wold decomposition of a single isometry.

Theorem 2.2 (Cf. [14, Theorem 1.2])

Let $S = [S_1, \ldots , S_m]$ be a row isometry on H. Then H decomposes into two S-reducing subspaces

$$ \begin{align*} H = H_u \oplus H_{s}, \end{align*} $$

such that $S|_{H_u}$ is a row unitary and $S|_{H_{s}}$ is an m-shift.

Furthermore,

$$ \begin{align*}H_u = \bigcap_{k\geq 0} \bigoplus_{|w| = k} S_w H, \end{align*} $$

and

$$ \begin{align*} H_{s} = \bigoplus_{w\in \mathbb{F}_n^+}S_w M, \end{align*} $$

where $M = \bigcap _{i=1}^n \ker (S_i^*)$ .

Definition 2.2 When S is a row isometry on a Hilbert space H, the decomposition $H = H_s \oplus H_u$ described in Theorem 2.2 is called the Wold decomposition of S.

2.1 The Lebesgue–Wold decomposition

Just as a unitary can be decomposed into its singular and absolutely continuous parts, a row unitary can be decomposed further. We will briefly summarize these results now, drawing largely from [2, 10].

Let $L = [L_1, \ldots , L_m]$ be the m-shift described in Example 2.1. Denote by $A_m$ and $\mathcal {L}_m$ the following two algebras:

The algebra $A_m$ is called the noncommutative disk algebra, and the algebra $\mathcal {L}_m$ is called the noncommutative analytic Toeplitz algebra.

Let $S = [S_1, \ldots , S_m]$ be a row isometry on a Hilbert space H. The free semigroup algebra generated by S is the algebra

Popescu [16] observed that the unital, norm-closed algebra generated by $S_1, \ldots , S_m$ is completely isometrically isomorphic to the noncommutative disk algebra $A_m$ . The free semigroup algebra $\mathcal {S}$ , however, can be very different from $\mathcal {L}_m$ .

Definition 2.3 Let $S = [S_1,\ldots ,S_m]$ be a row isometry on a Hilbert space H with $m\geq 2$ .

1. (i) There is a completely isometric isomorphism

   $$ \begin{align*} \Phi \colon A_m \rightarrow \operatorname{\mathrm{Alg}}\{I, S_1, \ldots, S_m\overline{\}}^{\|\cdot\|}, \end{align*} $$

   such that $\Phi (L_i) = S_i$ for $1\leq i \leq m$ . The row isometry S is absolutely continuous if $\Phi $ extends to a weak- $^*$ continuous representation of $\mathcal {L}_m$ .

2. (ii) The row isometry S is singular if S has no absolutely continuous restriction to an invariant subspace.

3. (iii) The row isometry S is of dilation type if it has no singular and no absolutely continuous summands.

Remark 2.3

1. (i) Absolute continuity for row isometries was introduced by Davidson, Li, and Pitts [3]. We refer the reader to [3, Section 2] or [10, Section 2] for details on why Definition 2.3 (i) generalizes the notion of a unitary with absolutely continuous spectral measure.

2. (ii) By [10, Theorem 5.1], a row isometry $S = [S_1,\ldots , S_m]$ , with $m\geq 2$ , is singular if and only if the free semigroup algebra $\mathcal {S}$ generated by S is a von Neumann algebra. Read [17] gave the first example of a self-adjoint free semigroup algebra, by showing that $B(H)$ is a free semigroup algebra (see also [1]).

3. (iii) The name “dilation type” is justified in [10, Proposition 6.2]. If S is a row isometry of dilation type on H, then there is a minimal subspace $V \subseteq H$ such that V is invariant for each $S_i^*$ , $1\leq i \leq m$ , and the restriction of S to $V^\perp $ is an m-shift. In which case, S is the minimal isometric dilation of the compression of S to V. In particular, if $K = (V + \sum _{i=1}^m S_i V) \ominus V$ , then $H = V \oplus \bigoplus _{w\in \mathbb {F}_m^+}S_w K.$

We can now describe the Lebesgue–Wold decomposition of a row isometry, due to Kennedy [10].

Theorem 2.4 (Cf. [10, Theorem 6.5])

If S is a row isometry on H, then H decomposes into four spaces which reduce S:

$$ \begin{align*} H = H_{\mathrm{abs}} \oplus H_{\mathrm{sing}} \oplus H_{\mathrm{dil}} \oplus H_{s}, \end{align*} $$

where $H_{\mathrm{abs}} \oplus H_{\mathrm{sing}} \oplus H_{\mathrm{dil}}$ and $H_s$ are the unitary and m-shift parts of the Wold decomposition, respectively. Furthermore, we have the following properties:

1. (i) $S|_{H_{\mathrm{abs}}}$ is absolutely continuous.

2. (ii) $S|_{H_{\mathrm{sing}}}$ is singular.

3. (iii) $S|_{H_{\mathrm{dil}}}$ is of dilation type.

This decomposition is unique.

Kennedy [10, Theorem 4.16] gives another characterization of absolute continuity. Let $S=[S_1,\ldots , S_m]$ be a row isometry with $m\geq 2$ , and let $\mathcal {S}$ be the free semigroup algebra generated by S. Then S is absolutely continuous if and only if $\mathcal {S}$ is isomorphic to $\mathcal {L}_m$ . This characterization answered a question asked in [3].

The property of $\mathcal {S}$ being isomorphic to $\mathcal {L}_m$ plays an important role in the work of Davidson, Katsoulis, and Pitts [2] in describing the structure of free semigroup algebras. We summarize the results which will be relevant to us now. Note that what we are calling “absolutely continuous” was called “type L” in [2]. The equivalence of the terms is due to the aforementioned work of Kennedy [10].

Theorem 2.5 (Cf. [2, Theorem 2.6])

Let $S = [S_1,\ldots ,S_m]$ be a row isometry on a Hilbert space H with $m\geq 2$ . Let $\mathcal {S}$ be the free semigroup algebra generated by S. There is a largest projection P in $\mathcal {S}$ such that $P\mathcal {S} P$ is self-adjoint. Furthermore, the following are satisfied:

1. (i) $PH$ is $S^*$ -invariant.

2. (ii) The restriction of S to $P^\perp H$ is an absolutely continuous row isometry.

Definition 2.4 Let S be a row isometry, and let P be the projection described in Theorem 2.5. Then P is called structure projection for S.

Let $S = [S_1, \ldots , S_m]$ be a row isometry on H, with $H = H_{\mathrm{abs}} \oplus H_{\mathrm{sing}} \oplus H_{\mathrm{dil}} \oplus H_{s}$ being the Lebesgue–Wold decomposition. Furthermore, write $H_{\mathrm{dil}} = V \oplus \bigoplus _{w\in \mathbb {F}_m^+} S_w K$ , as described in Remark 2.3(iii). It follows from Theorems 2.4 and 2.5 that

$$ \begin{align*} PH = H_{\mathrm{sing}} \oplus V. \end{align*} $$

3 Słociński–Wold decompositions for $\theta $ -commuting row isometries

Definition 3.1 Let $A=[A_1,\ldots ,A_m]$ and $B=[B_1,\ldots ,B_n]$ be two row operators on a Hilbert space H, and let $\theta \in S_{m\times n}$ be a permutation. We say that A and $B \ \theta $ -commute if

$$ \begin{align*} A_i B_j =B_{j'}A_{i'} \end{align*} $$

when $\theta (i,j) = (i',j')$ . When $\theta $ is the identity permutation, we will say that A and B commute.

If A and B are $\theta $ -commuting row operators which further satisfy

$$ \begin{align*} B_j^* A_i &= \sum_{\theta(k,j) = (i,j_k)} A_k B_{j_k}^* \text{ and}\\ A_i^* B_j &= \sum_{\theta(i,k) = (i_k, j)}B_k A_{i_k}^*, \end{align*} $$

we say that A and $B \ \theta $ -doubly commute.

The following lemma is proved by repeated applications of the commutation rule from $\theta $ . It will be used liberally in the sequel.

Lemma 3.1 Let $A=[A_1,\ldots ,A_m]$ and $B=[B_1,\ldots ,B_n]$ be $\theta $ -commuting row operators. For each $k, l \geq 1$ , $\theta $ determines a permutation $\theta _{k,l} \in S_{m^k \times n^l}$ so that

$$ \begin{align*} A_u B_w = B_{w'} A_{u'} \end{align*} $$

when $\theta _{k,l}(u,w) = (u',w')$ .

Any $2$ -graph with a single vertex, in the sense of [11], is uniquely determined by a single permutation. Thus, two $\theta $ -commuting row contractions A and B determine a contractive representation of single vertex $2$ -graph. This is the perspective $\theta $ -commuting row operators are studied from in, e.g., [4, 5, 7].

Definition 3.2 Let $S=[S_1,\ldots , S_m]$ and $T=[T_1,\ldots ,T_n]$ be $\theta $ -commuting row isometries on a Hilbert space H. We say that S and T have a Słociński–Wold decomposition if H decomposes into

$$ \begin{align*} H = H_{uu} \oplus H_{us} \oplus H_{su} \oplus H_{ss}, \end{align*} $$

where $H_{uu}$ , $H_{us}$ , $H_{su}$ , and $H_{ss}$ are both S-reducing and T-reducing subspaces satisfying:

1. (i) $S|_{H_{uu}}$ and $T|_{H_{uu}}$ are both row unitaries.

2. (ii) $S|_{H_{us}}$ is a row unitary, and $T|_{H_{us}}$ is an n-shift.

3. (iii) $S|_{H_{su}}$ is an m-shift, and $T|_{H_{su}}$ is a row unitary.

4. (iv) $S|_{H_{ss}}$ is an m-shift, and $T|_{H_{ss}}$ is an n-shift.

The following general lemma will be used throughout our analysis.

Lemma 3.2 $S=[S_1,\ldots ,S_m]$ is a row isometry which $\theta $ -commutes with a row operator $A = [A_1,\ldots , A_l]$ . Let $H = H_u \oplus H_s$ be the Wold decomposition of S. Then $H_u$ is A-invariant.

Proof Take $h \in H_u$ and fix $k \geq 0$ . Since S is a row unitary on $H_u$ ,

$$ \begin{align*} h = \sum_{|w|=k} S_w S_w^* h. \end{align*} $$

Choose an $A_i$ , $1\leq i \leq l$ . For each w with $|w|=k$ , there is a $w'$ with $|w'|=k$ , and $i_w$ with $1\leq i_w \leq l$ so that $A_iS_w = S_{w'}A_{i_w}$ . Thus,

$$ \begin{align*} A_i h & = A_i \sum_{|w|=k} S_w S_w^* h \\ & = \sum_{|w|=k}S_{w'}A_{i_w}S_w^*h \in \sum_{|w|=k} S_w H. \end{align*} $$

Since this holds for all $k\geq 0$ , $A_i H_u \subseteq H_u$ by Theorem 2.2.

We can now give a general statement on the existence of Słociński–Wold decompositions. The case when $m = n = 1$ is covered in [19, Proposition 3].

Proposition 3.3 Let $S = [S_1, \ldots , S_m]$ and $T = [T_1,\ldots ,T_n]$ be $\theta $ -commuting row isometries on H. Then S and T have a Słociński–Wold decomposition if and only if:

1. (i) if $H = H_u^S \oplus H_s^S$ is the Wold decomposition of S, then $H_u^S$ reduces T; and

2. (ii) if $H_u^S = H_u^T \oplus H_s^T$ is the Wold decomposition of $T|_{H_s^S}$ , then $H_u^T$ reduces S.

Proof If S and T have a Słociński–Wold decomposition, then conditions (i) and (ii) are clearly satisfied.

Suppose now that conditions (i) and (ii) are satisfied. Let $H = H_u^S \oplus H_s^S$ be the Wold decomposition for S. Let $H_u^S = K_u^T \oplus K_s^T$ be the Wold decomposition of $H_u^S$ from the restriction of T to $H_u^S$ . By Lemma 3.2, $K_u^T$ is S-invariant. Take any $1\leq i \leq m$ , and $h \in K_u^T$ . Recall, by Lemma 3.1, for each $k\geq 1$ , there is a permutation $\theta _{1,k}$ on $S_{m \times n^k}$ so that for $1\leq i \leq m$ and $w \in \mathbb {F}_n^+$ , $S_i T_w = T_{w'}S_{i'}$ when $\theta _{1,k}(i,w) = (i',w')$ . Hence, for every $k\geq 1$ ,

$$ \begin{align*} S_i^*h & = S_i^* \sum_{|w|=k}T_wT_w^* h \\ & = \sum_{|w|=k}S_i^* T_wT_w^* h \\ &= \sum_{|w|=k}\sum_{l = 1}^m S_i^* T_w S_l S_l^* T_w^* h \\ &= \sum_{|w|=k}\sum_{\theta_{1,k}(i,w_i) = (l,w)}T_{w_i}S_l^* T_w^* h \\ & \in \bigoplus_{|w|=k}T_w H_u^S, \end{align*} $$

where the fact that S is a row unitary on $H_u^S$ is used in the third equality. It follows from Theorem 2.2 that $S_i^*h \in K_u^T$ . Hence, $K_u^T$ is S-reducing.

Letting $H_s^S = H_u^T \oplus H_s^T$ be the Wold decomposition of $T|_{H_u^S}$ , we have that $H_{uu} = K_u^T$ , $H_{us} = K_s^T$ , $H_{su} = H_u^T$ , and $H_{ss} = H_s^T$ gives the desired Słociński–Wold decomposition.

Skalski and Zacharias studied Wold decompositions of isometric representations of product systems of C $^*$ -correspondences [18]. The following is a special case of one of their results.

Theorem 3.4 (Cf. [18, Theorem 2.4])

If S and T are $\theta $ -double commuting row isometries, then they have a Słociński–Wold decomposition.

Proof Let $H = H^S_u \oplus H^S_s$ be the Wold decomposition of H from S. We will show that $H_u^S$ is T-reducing. Lemma 3.2 gives that $H_u^S$ is T-invariant, so it only remains to show that $H_u^S$ is $T^*$ -invariant. Take $1\leq j \leq n$ and $h \in H_u^S$ . Using the condition that S and $T \ \theta $ -doubly commute and that S is a row unitary on $H_u^S$ , we have, for every $k \geq K$ ,

$$ \begin{align*} T_j^* h &= \sum_{|w|=k} T_j^* S_w S_w^*h \\ & = \sum_{\theta_{k,1}(w_k,j) = (w,j_w)} S_{w_k}T_{j_w}^* S_w^* h\\ &\in \sum_{|w|=k}S_w H. \end{align*} $$

Thus, $T_j^* h \in H_u^S$ by Lemma 2.2.

Now, let $H_s^S = H_u^T \oplus H_s^T$ be the Wold decomposition of $T|_{H_s^S}$ . The same calculation as above, with the roles of S and T swapped, shows that $H_u^T$ is S-reducing. Thus, S and T have a Słociński–Wold decomposition by Proposition 3.3.

Remark 3.5 As described in [18], the Słociński–Wold decomposition for $\theta $ -doubly commuting row isometries has additional structure on the shift part $H_{ss}$ . On $H_{ss}$ , S and T are not just both (m and n) shifts. The operators S and T work as shifts together, giving an ampliation of the left-regular representation of the unital semigroup

$$ \begin{align*} F_\theta^+ = \langle i_1,\ldots,i_m, j_1,\ldots, j_n \colon i_k j_l = j'i' \text{ when } \theta(i_k,j_l)=(i',l') \rangle. \end{align*} $$

Explicitly, if $M = \bigcap _{i=1}^m \ker S_i^* \cap \bigcap _{j=1}^n \ker T_j^*$ , then

$$ \begin{align*} H_{ss} = \bigoplus_{u \in \mathbb{F}_m^+,\ w\in \mathbb{F}_n^+} S_u T_w M. \end{align*} $$

Theorem 3.4 generalizes Theorem 3 of [19]. In the rest of this note, we will give analogues of Theorems 4 and 5 of [19] for $\theta $ -commuting row isometries. That is, we will give sufficient conditions for the existence of a Słociński–Wold decomposition for $\theta $ -commuting row isometries based on the Lebesgue decomposition of their unitary parts.

Lemma 3.6 Let $S=[S_1,\ldots ,S_m]$ be a row isometry on H with $m\geq 2$ , and let P be the structure projection for S. If $T = [T_1,\ldots , T_n]$ is a row isometry on H which $\theta $ -commutes with S. Then $PH$ is $T^*$ -invariant.

Proof By Theorem 2.2, S is absolutely continuous on $P^\perp H$ . Thus, by [10, Corollary 4.17], $P^\perp H$ is spanned by wandering vectors for S. Recall that a vector $h \in H$ is wandering for S if $\langle S_w h, h\rangle = 0$ for all $w\in \mathbb {F}_m^+$ , $w \neq \emptyset $ . Let h be a wandering vector for S. Then, for any $1\leq j \leq n$ and $w \in \mathbb {F}_n^+$ , $|w|\geq 1$ , we have

$$ \begin{align*} \langle S_w T_j h, T_j h\rangle &= \langle S_{w'}h, T_{j'}^*T_j h\rangle, \end{align*} $$

where $w'$ and $j'$ satisfy $S_w T_j = T_{j'}S_{w'}$ . If $j'\neq j$ , then $T_{j'}^* T_j = 0$ , in which case $\langle S_w T_j h, T_j h\rangle = 0$ . If $j' = j$ , then

$$ \begin{align*} \langle S_w T_j h, T_j h\rangle = \langle S_{w'}h,h\rangle = 0, \end{align*} $$

since h is wandering for S and $|w'|=|w|\geq 1$ . Hence, $T_jh$ is wandering for S, and so $T_j h \in P^{\perp } H$ . It follows that $T_jP^{\perp } H \subseteq P^\perp H$ , and hence $PH$ is $T^*$ -invariant.

Let V be an isometry on a Hilbert space H, and let $N \in B(H)$ be an operator commuting with V. Let $H = H_{\mathrm{abs}} \oplus H_{\mathrm{sing}} \oplus H_s$ be the Lebesgue–Wold decomposition of V. It then follows from [12, Theorem 2.1] that $H_{\mathrm{sing}}$ reduces N. Thus, if $H_{\mathrm{abs}} = \{0\}$ , the unitary part of V reduces N. In Proposition 3.7, we show that if S and T are $\theta $ -commuting row isometries and the unitary part of S is singular, then the Wold decomposition of S reduces T.

Proposition 3.7 Let $S=[S_1,\ldots ,S_m]$ and $T = [T_1,\ldots , T_n]$ be $\theta $ -commuting row isometries on H. Let $H = H_u \oplus H_s$ be the Wold decomposition for S. If the unitary part of S is singular, then $H_u$ reduces T

Proof When $m = 1$ , the result follows from [12, Theorem 2.1] (see [19, Remark 2]). Otherwise, we have $H_u = PH$ where P is the structure projection for S. The result follows from Lemmas 3.2 and 3.6.

We now give a row-isometry analog of [19, Theorem 4].

Theorem 3.8 Let $S=[S_1,\ldots ,S_m]$ and $T=[T_1,\ldots ,T_n]$ be $\theta $ -commuting row isometries on a Hilbert space H. Furthermore, suppose that the unitary parts of S and T are singular. Then S and T have a Słociński–Wold decomposition.

Proof The result follows immediately from Propositions 3.3 and 3.7.

The following lemma generalizes [19, Lemma 2] to row isometries. It is notable that the conditions are less restrictive for the row-isometry case than they are in single-isometry case dealt with in [19].

Lemma 3.9 Let S be an m-shift of finite multiplicity on a Hilbert space H. Let $T = [T_1,\ldots ,T_n]$ be a row unitary on H which $\theta $ -commutes with S. If

1. (1) $n\geq 2$ , or

2. (2) $n = 1$ and T has empty point spectrum,

then $H = \{0\}$ .

Proof Let $L = \bigcap _{i=1}^m \ker S_i^*$ . By assumption, L is finite-dimensional. Since T and $S \ \theta $ -commute, it is clear that L is $T^*$ -invariant. As T is a row unitary, if $h \in L$ and $1\leq i \leq m$ , we have that

$$ \begin{align*} S_i^* T_j h = \sum_{k=1}^n T_kT_k^* S_i^* T_j h = \sum_{\theta(i,k) = (i_k, j)}T_k S_{i_k}^* h = 0,\end{align*} $$

and so L is T-reducing.

If $n\geq 2$ , then $T_1|_L,\ldots , T_n|_L$ are isometries with pairwise orthogonal finite-dimensional ranges. If $n=1$ , then $T|_L$ is a unitary on a finite-dimensional space and so has an eigenvalue. In either case, we see that we must have $L = \{0\}$ and hence $H = \{0\}$ .

We end with the following generalization of [19, Theorem 5].

Theorem 3.10 Let $S=[S_1,\ldots ,S_m]$ and $T=[T_1,\ldots ,T_n]$ be $\theta $ -commuting row isometries on a Hilbert space H. Assume that the unitary part of S is singular, and that the shift part of S has finite multiplicity, then S and T have a Słociński–Wold decomposition if

1. (i) $n\geq 2$ ; or

2. (ii) $n = 1$ and $\theta $ is the identity permutation.

Proof Let $H = H_u^S \oplus H_s^S$ . As S has only singular unitary part, $H_u^S$ reduces T by Proposition 3.7. Let $H_s^S = K_u^T \oplus K_s^T$ be the Wold decomposition of the restriction of T to $H_s^S$ . Lemma 3.2 says that $K_u^T$ is S-invariant. As S is an m-shift of finite multiplicity on $H_s^S$ , the restriction of S to $K_u^T$ is an m-shift of finite multiplicity. When $m=1$ , this is [9, Lemma 4]; when $m\geq 2$ , it follows from [15, Theorem 3.1] and [15, Theorem 3.2].

When $n \geq 2$ , it follows from Lemma 3.9 that $K_u^T = \{0\}$ and hence S and T have a Słociński–Wold decomposition by Proposition 3.3. When $n=1$ and T is an isometry commuting with each $S_i$ , the proof follows as in [19, Theorem 4].