Abstract
In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators \(B_{1},\ldots,B_{n}\), where the n-tuple \((B_{1},\ldots,B_{n})\) is the universal model associated with the abstract noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}\).
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1 Introduction
In the last fifty years, the study of the closed operator unit ball
has generated the celebrated Sz.-Nagy–Foiaş theory of contractions on Hilbert spaces. This research has evolved into a well-developed theory, which plays an important role in modern functional analysis. In 1963, Sz.-Nagy and Foiaş obtained an effective \(H^{\infty}\)-functional calculus for completely nonunitary contractions on Hilbert spaces based on the existence of a unitary dilation of a contraction T (see [33]). An important application of this functional calculus to the theory of contraction semigroups has also been given in Foiaş [5]. Moreover, the characteristic function of a contraction T appears as the operator-valued analytic function corresponding to a certain orthogonal projection in the space of the minimal unitary dilation of T. This yields a functional model for T, which is a useful tool for analyzing the structure of contractions.
In the multivariable case, the study of the closed operator unit n-ball
has generated a noncommutative analogue of Sz.-Nagy–Foiaş theory (see [2–4, 6–8], and more recently [1, 11, 34]). In particular, Popescu developed a theory of holomorphic functions in several noncommuting variables and provided a framework for the study of arbitrary n-tuples of operators. A free analytic functional calculus was introduced and studied in connection with Hausdorff derivations, noncommutative Cauchy and Poisson transforms, and von Neumann inequalities (see [15, 16, 18, 20–23, 26, 29, 30]). Moreover, we remark the work of Helton, McCullough, and Vinnikov on symmetric noncommutative polynomials (see [9, 10]). We should also remark that, in recent years, many results concerning the theory of row contractions were extended by Muhly and Solel ([12–14]) to representations of tensor algebras over \(C^{*}\)-correspondences and Hardy algebras.
In [28], Popescu developed an operator model theory for pure n-tuples of operators in noncommutative domains \(\mathbb{D}_{f,\varphi }(\mathcal{H})\subset B(\mathcal{H})^{n}\) generated by positive regular free holomorphic functions f and certain classes of n-tuples \(\varphi=(\varphi _{1},\ldots,\varphi_{n})\) of formal power series in noncommutative indeterminates \(Z_{1},\ldots,Z_{n}\). An important role in his study was played by noncommutative Poisson transforms. Using these transforms, he proved that each abstract noncommutative domain \(\mathbb{D}_{f,\varphi}\) has a universal model \((M_{Z_{1}},\ldots,M_{Z_{n}})\). Unlike the case of the ball \([B(\mathcal{H})^{n}]_{1}^{-}\), the operators \(M_{Z_{1}},\ldots,M_{Z_{n}}\) are not isometries and do not have orthogonal ranges in general, which leads to considerable technical difficulties in developing an operator model theory. Moreover, notice that the study of \(\mathbb{D}_{f,\varphi }(\mathcal{H})\) is closely related to the study of the operators \(M_{Z_{1}},\ldots,M_{Z_{n}}\), their joint invariant subspaces, and the representations of the algebras they generate: the noncommutative domain algebra \(\mathcal{A}(\mathbb{D}_{f,\varphi})\), the noncommutative Hardy algebra \(H^{\infty}(\mathbb{D}_{f,\varphi})\), and the \(C^{*}\)-algebra \(C^{*}(M_{Z_{1}},\ldots,M_{Z_{n}})\). Indeed, this noncommutative domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) has been studied in several particular cases. According to [22, 24] and [33], if \(f=Z\) and \(\varphi=Z\), then the corresponding domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) coincides with the closed operator unit ball \([B(\mathcal{H})]_{1}^{-}\), the study of which has generated Sz.-Nagy–Foiaş theory of contractions. If \(f=Z_{1}+\cdots+Z_{n}\) and \(\varphi=(Z_{1},\ldots,Z_{n})\), then the corresponding domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) coincides with the closed operator unit n-ball \([B(\mathcal {H})^{n}]_{1}^{-}\), the study of which has generated a free analogue of Sz.-Nagy–Foiaş theory. In particular, if \(\varphi=(Z_{1},\ldots ,Z_{n})\), then the corresponding domain \(\mathbb{D}_{f,\varphi}(\mathcal {H})\) coincides with the noncommutative Reinhardt domain \(\mathcal {D}_{f}(\mathcal{H})\), which was first studied by Popescu [24].
In this paper, we continue the research line of Popescu to develop an operator model theory for completely non-coisometric n-tuples of operators in noncommutative varieties \(\mathcal{V}_{f,\varphi,\mathcal {I}}(\mathcal{H})\). To present our results, we need some notation. Let \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\) be the algebra of all formal power series in noncommutative indeterminates \(Z_{1},\ldots,Z_{n}\) and complex coefficients. We denote by \(\mathbb{F}_{n}^{+}\) the unital free semigroup on n generators \(g_{1},\ldots ,g_{n}\) and the identity \(g_{0}\). The length of \(\alpha\in\mathbb{F}_{n}^{+}\) is defined by \(|\alpha|:=0\) if \(\alpha=g_{0}\) and \(|\alpha|:=k\) if \(\alpha =g_{i_{1}}\cdots g_{i_{k}}\), where \(i_{1},\ldots,i_{k}\in\{1,\ldots,n\}\). We set \(Z_{\alpha}:=Z_{i_{1}}\cdots Z_{i_{k}}\) and \(Z_{g_{0}}:= I\). If \(f\in\mathbf{S}[Z_{1},\ldots,Z_{n}]\) has the representation \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\) and the coefficients \(a_{\alpha}\in\mathbb{C}\) satisfy the conditions
\(a_{\alpha}\ge0\) for any \(\alpha\in\mathbb{F}_{n}^{+}\), \(a_{g_{0}}=0\), and \(a_{g_{i}}>0\), \(i=1,\ldots,n\), we say that f is a positive regular free holomorphic function. The number \(r(f)\) is called the radius of convergence of f.
Denote by \(\mathcal{M}_{f}\) the set of all n-tuples \(\varphi=(\varphi _{1},\ldots,\varphi_{n})\) of formal power series \(\varphi_{i}\in\mathbf {S}[Z_{1},\ldots, Z_{n}]\) with the model property (see Sect. 2). \(\mathcal{H}\) is a Hilbert space and \(B(\mathcal{H})\) is the algebra of all bounded linear operators on \(\mathcal{H}\). If \(X = (X_{1},\ldots, X_{n}) \in B(\mathcal{H})^{n}\), we denote \(X_{\alpha}:=X_{i_{1}}\cdots X_{i_{k}}\) if \(\alpha=g_{i_{1}}\cdots g_{i_{k}}\in\mathbb{F}_{n}^{+}\), and \(X_{g_{0}} := I_{\mathcal{H}}\). We introduce the noncommutative domain \(\mathbb {D}_{f,\varphi}(\mathcal{H})\) associated with \(f,\varphi\in\mathcal{M}_{f}\) and a Hilbert space \(\mathcal{H}\) and defined by
where \(\psi:=(\psi_{1},\ldots,\psi_{n})\) is the inverse of φ with respect to composition of formal power series, and the evaluations are well defined (see Sect. 2). We refer to \(\mathbb{D}_{f,\varphi }:=\{\mathbb{D}_{f,\varphi}(\mathcal{H}): \mathcal{H}\text{ is a Hilbert space}\}\) as the abstract noncommutative domain, and to \(\mathbb {D}_{f,\varphi}(\mathcal{H})\) as its representation on the Hilbert space \(\mathcal{H}\). We associate with each \(\mathbb{D}_{f,\varphi}\) a Hilbert space \(\mathbb {H}_{f}^{2}(\varphi)\) of formal power series in \(\mathbf{S}[Z_{1},\ldots, Z_{n}]\) with the property that the indeterminates \(Z_{1},\ldots, Z_{n}\) are in the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\) and each left multiplication operator \(M_{Z_{i}}:\mathbb{H}_{f}^{2}(\varphi )\to\mathbb{H}_{f}^{2}(\varphi)\) defined by
is a bounded multiplier of \(\mathbb{H}_{f}^{2}(\varphi)\). Similarly, each right multiplication operator \(R_{Z_{i}}:\mathbb{H}_{f}^{2}(\varphi)\to\mathbb {H}_{f}^{2}(\varphi)\) defined by
is also a bounded multiplier of \(\mathbb{H}_{f}^{2}(\varphi)\).
Let \(\mathcal{I}\neq H^{\infty}(\mathbb{D}_{f,\varphi})\) be a WOT-closed two-sided ideal of the noncommutative Hardy algebra \(H^{\infty}(\mathbb{D}_{f,\varphi})\), where \(H^{\infty}(\mathbb {D}_{f,\varphi})\) is the WOT-closure of all noncommutative polynomials in \(M_{Z_{1}},\ldots,M_{Z_{n}}\) and the identity. Now we define the noncommutative variety
Denote by \(H^{\infty}(\mathcal{V}_{f,\varphi,\mathcal{I}})\) the WOT-closed algebra generated by the constrained weighted shifts \(B_{i}:=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\) for \(i=1,\ldots,n\) and the identity, where
Similarly, denote by \(R^{\infty}(\mathcal{V}_{f,\varphi,\mathcal{I}})\) the WOT-closed algebra generated by the constrained weighted shifts \(C_{i}:=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}R_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\) for \(i=1,\ldots,n\) and the identity.
In Sect. 2, we collect some notation and preliminaries which are needed in the sequel. In Sect. 3, we obtain a factorization result for the constrained characteristic function, namely
where \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) is the constrained characteristic function and \(K_{f,\varphi,T}^{(\mathcal{I})}\) is the corresponding constrained Poisson kernel. Moreover, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. Applying this result, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements. Indeed, this result can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions.
In Sect. 4, we prove a Sarason-type commutant lifting theorem. As an application, we obtain the Nevanlinna–Pick-type interpolation result in our setting. We show that if \(\lambda_{1},\ldots,\lambda_{k}\) are k distinct points in the strict noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal {I}}^{<}(\mathbb{C})\) and \(A_{1},\ldots,A_{k}\in B(\mathcal{K})\), then there exists \(\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}(\mathcal{V}_{f,\varphi ,\mathcal{I}})\mathrel{\overline{\otimes}}B(\mathcal{K})\) such that
if and only if the operator matrix
is positive semidefinite, where
Moreover, we provide a Beurling-type characterization of the joint invariant subspaces under the constrained weighted shifts \(B_{1},\ldots ,B_{n}\). More precisely, a subspace \(\mathcal{M}\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}\) is invariant under \(B_{i}\otimes I_{\mathcal{K}}\), \(i=1,\ldots,n\), if and only if there are a Hilbert space \(\mathcal{G}\) and an inner multi-analytic operator
with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\) such that
2 Preliminaries
In this section we collect some notation and preliminaries which are needed in the sequel. For more information, we refer to [24, 27] and [28].
2.1 Weighted Fock space
Let \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\), \(a_{\alpha}\in \mathbb{C}\), be a positive regular free holomorphic function. Define the noncommutative domain
where the convergence of the series is in the weak operator topology. Define the strict noncommutative domain
where the convergence is in the weak operator topology. Now, we define
We introduce an inner product on the algebra of noncommutative polynomials \(\mathbb{C}[Z_{1},\ldots,Z_{n}]\) by setting
Let \(\mathcal{F}_{f}^{2}\) be the completion of \(\mathbb{C}[Z_{1},\ldots,Z_{n}]\) in this inner product. Notice that the elements of \(\mathcal{F}_{f}^{2}\) are formal power series \(\zeta\in\mathbf{S}[Z_{1},\ldots,Z_{n}]\) of the form \(\zeta=\sum_{\alpha \in\mathbb{F}_{n}^{+}}c_{\alpha}Z_{\alpha}\), where
Indeed, \(\mathcal{F}_{f}^{2}\) is a weighted Fock space on n generators. For each \(i=1,\ldots,n\), we define the left multiplication operator \(V_{i}:\mathcal{F}_{f}^{2}\to\mathcal{F}_{f}^{2}\) by setting \(V_{i}\zeta:=Z_{i}\zeta\). Notice that \((V_{1},\ldots,V_{n})\) is in the noncommutative domain \(\mathcal {D}_{f}(\mathcal{F}_{f}^{2})\), and
where \(P_{\mathbb{C}}\) is the orthogonal projection from \(\mathcal {F}_{f}^{2}\) onto \(\mathbb{C}\).
Let \(\mathcal{F}_{f}^{\infty}\) be the set of all \(\zeta\in\mathcal{F}_{f}^{2}\) with the property that
Notice that \(\mathcal{F}_{f}^{\infty}\) is a Banach algebra with respect to the norm \(\|\cdot\|_{\infty}\). Let \(\zeta=\sum_{\beta\in\mathbb {F}_{n}^{+}}c_{\beta}Z_{\beta}\) be a formal power series with the property that \(\sum_{\beta\in\mathbb {F}_{n}^{+}}|c_{\beta}|^{2} \frac{1}{b_{\beta}}<\infty\), where the coefficients \(b_{\beta}\), \(\beta\in\mathbb{F}_{n}^{+}\), are given by relation (2.1). One can see that \(\sum_{\beta\in \mathbb{F}_{n}^{+}}c_{\beta}V_{\beta}(p)\in\mathcal{F}_{f}^{2}\) for any \(p\in \mathbb{C}[Z_{1},\ldots,Z_{n}]\). Moreover, \(\zeta\in\mathcal{F}_{f}^{\infty}\) if and only if
In this case, there is a unique bounded operator acting on \(\mathcal {F}_{f}^{2}\), which we denote by \(\zeta(V_{1},\ldots,V_{n})\), such that
We call the series \(\sum_{\beta\in\mathbb{F}_{n}^{+}}c_{\beta}V_{\beta}\) the Fourier representation of \(\zeta(V_{1},\ldots,V_{n})\). The set of all operators \(\varphi(V_{1},\ldots,V_{n})\in B(\mathcal {F}_{f}^{2})\) satisfying the above-mentioned properties is denoted by \(F^{\infty}(\mathcal{D}_{f})\).
We consider the full Fock space of \(H_{n}\) defined by
where \(H_{n}^{\otimes m}\) is the Hilbert tensor product of m copies of \(H_{n}\). We denote \(e_{\alpha}:=e_{i_{1}}\otimes\cdots\otimes e_{i_{k}}\) if \(\alpha=g_{i_{1}}\cdots g_{i_{k}}\), where \(i_{1},\ldots,i_{k}\in\{1,\ldots ,n\}\), and \(e_{g_{0}}:=1\). Consider \(\varOmega:F^{2}(H_{n})\to\mathcal{F}_{f}^{2}\) to be the unitary operator defined by \(\varOmega(e_{\alpha}):=\sqrt{b_{\alpha}}Z_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), where the coefficients \(b_{\alpha}\) are given by relation (2.1). We remark that \(\varOmega ^{-1}V_{i}\varOmega=W_{i}\), \(i=1,\ldots,n\), where \((W_{1},\ldots,W_{n})\) is the n-tuple of weighted shifts on \(F^{2}(H_{n})\), which was introduced in [24]. Using the results from [24], we know that \(F^{\infty}(\mathcal{D}_{f})\) is the WOT-closure (resp. SOT-closure, \(w^{*}\)-closure) of all polynomials in \(V_{1},\ldots ,V_{n}\) and the identity. The noncommutative domain algebra \(\mathcal {A}(\mathcal{D}_{f})\) is the norm-closure of all polynomials in \(V_{1},\ldots,V_{n}\) and the identity.
2.2 Noncommutative domain
We say that an n-tuple \(p=(p_{1},\ldots,p_{n})\) of polynomials is invertible with respect to composition if there exists an n-tuple \(q=(q_{1},\ldots,q_{n})\) of polynomials such that \(p\circ q=q\circ p=\mathit{id}\). In this case, we say that p has property (\(\mathcal{A}\)). In what follows, we provide an example. If
then \(p=(p_{1},p_{2},p_{3})\) is invertible with respect to composition, i.e., there exists \(q=(q_{1},q_{2},q_{3})\) such that \(p\circ q=q\circ p=\mathit{id}\), where
This shows that p has property (\(\mathcal{A}\)).
Let \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\) be a positive regular free holomorphic function, and let \(p=(p_{1},\ldots ,p_{n})\) be an n-tuple of noncommutative polynomials with property (\(\mathcal{A}\)). We introduce an inner product by setting
Let \(\mathbb{H}_{f}^{2}(p)\) be the completion of the linear space \(\bigvee\{ p_{\alpha}\}_{\alpha\in\mathbb{F}_{n}^{+}}\) with respect to this inner product.
Consider an n-tuple of formal power series \(\varphi=(\varphi _{1},\ldots,\varphi_{n})\) in indeterminates \(Z_{1},\ldots,Z_{n}\) with the property that the Jacobian
where
and \(i,j=1,\ldots,n\). Due to Theorem 1.2 from [25], the set \(\{\varphi_{\alpha}\}_{\alpha \in\mathbb{F}_{n}^{+}}\) (where \(\varphi_{0}:=I\)) is linearly independent in \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\). We introduce an inner product on the linear span of \(\{\varphi_{\alpha}\} _{\alpha\in\mathbb{F}_{n}^{+}}\) by setting
where the coefficients \(b_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), are given by relation (2.1). Let \(\mathbb{H}_{f}^{2}(\varphi)\) be the completion of the linear space \(\bigvee\{\varphi_{\alpha}\}_{\alpha\in\mathbb{F}_{n}^{+}}\) with respect to this inner product. Assume now that \(\varphi(0)=0\). Theorem 1.3 from [25] shows that φ is not a right zero divisor with respect to composition, i.e., there is no nonzero power series χ in \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\) such that \(\chi\circ\varphi=0\). Consequently, the elements of \(\mathbb{H}_{f}^{2}(\varphi)\) can be seen as a formal power series in \(\mathbf{S}[Z_{1},\ldots,Z_{n}]\) of the form \(\sum_{\alpha\in\mathbb{F}_{n}^{+}}c_{\alpha}\varphi_{\alpha}\), where \(\sum_{\alpha\in\mathbb{F}_{n}^{+}}\frac{1}{b_{\alpha}}|c_{\alpha}|^{2}<\infty\).
To introduce the class of n-tuples of formal power series with property \((\mathcal{S})\), we need some preliminaries. Let \(\chi=\sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}Z_{\alpha}\) be a formal power series in indeterminates \(Z_{1},\ldots,Z_{n}\). We denote by \(\mathcal {C}_{\chi}(\mathcal{H})\) (resp. \(\mathcal{C}_{\chi}^{\mathrm{SOT}} (\mathcal{H})\)) the set of all \(Y:=(Y_{1},\ldots,Y_{n})\in B(\mathcal{H})^{n}\) such that the series \(\chi (Y_{1},\ldots,Y_{n}):= \sum_{k=0}^{\infty}\sum_{|\alpha|=k}c_{\alpha}Y_{\alpha}\) is norm (resp. SOT) convergent. These sets are called sets of norm (resp. SOT) convergence for the power series χ. We also introduce the set \(\mathcal{C}_{\chi}^{\mathrm{rad}} (\mathcal{H})\) of all \(Y:=(Y_{1},\ldots,Y_{n})\in B(\mathcal{H})^{n}\) such that there exists \(\delta \in(0,1)\) with the property that \(rY\in\mathcal{C}_{\chi}(\mathcal{H})\) for any \(r\in (\delta,1)\) and
exists.
Definition 2.1
(see [28])
Let \(\varphi=(\varphi_{1},\ldots,\varphi_{n})\) be an n-tuple of formal power series in \(Z_{1},\ldots,Z_{n}\) such that \(\varphi(0)=0\). We say that φ has property (\(\mathcal{S}\)) if the following conditions hold:
- (\(\mathcal{S}_{1}\)):
The radius of convergence of φ, i.e., \(r(\varphi):=\min_{i=1,\ldots,n}r(\varphi_{i})\), is strictly positive and det \(J_{\varphi}(0)\neq0\).
- (\(\mathcal{S}_{2}\)):
The indeterminates \(Z_{1},\ldots,Z_{n}\) are in the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\) and each multiplication operator \(M_{Z_{i}}:\mathbb{H}_{f}^{2}(\varphi)\to\mathbb{H}_{f}^{2}(\varphi)\) defined by
$$M_{Z_{i}}\zeta:=Z_{i}\zeta,\quad\zeta\in \mathbb{H}_{f}^{2}(\varphi), $$is a bounded multiplier of \(\mathbb{H}_{f}^{2}(\varphi)\).
- (\(\mathcal{S}_{3}\)):
The multiplication operators \(M_{\varphi _{j}}:\mathbb{H}_{f}^{2}(\varphi)\to\mathbb{H}_{f}^{2}(\varphi)\), \(M_{\varphi _{j}}\chi=\varphi_{j}\chi\), satisfy the equations
$$M_{\varphi_{j}}=\varphi_{j}(M_{Z_{1}}, \ldots,M_{Z_{n}}),\quad j=1,\ldots,n, $$where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the convergence set \(\mathcal{C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal {C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\).
Let \(U:\mathbb{H}_{f}^{2}(\varphi)\to\mathcal{F}_{f}^{2}\) be the unitary operator defined by \(U(\varphi_{\alpha}):=Z_{\alpha}\), \(\alpha \in\mathbb{F}_{n}^{+}\). According to the proof of Lemma 1.2 from [28], we have
Throughout this paper, unless otherwise specified, we assume that \(\varphi=(\varphi_{1},\ldots, \varphi_{n})\) is either an n-tuple of noncommutative polynomials with property (\(\mathcal{A}\)) or an n-tuple of formal power series with \(\varphi(0)=0\) and property (\(\mathcal{S}\)). In this case, we say that φ has the model property.
Definition 2.2
Let \(\varphi=(\varphi_{1},\ldots,\varphi_{n})\) be an n-tuple of formal power series with model property, and let \(\psi=(\psi_{1},\ldots,\psi_{n})\) be the n-tuple of power series which is the inverse of \(\varphi =(\varphi_{1},\ldots,\varphi_{n})\) with respect to composition. Assume that \(\psi_{i}\) has the representation
where the sequence \(\{c_{\alpha}^{(i)}\}_{\alpha\in\mathbb{F}_{n}^{+}}\) is uniquely determined by the condition \(\psi\circ\varphi=\mathit{id}\). We say that an n-tuple of operators \(X=(X_{1},\ldots,X_{n})\in B(\mathcal {H})^{n}\) satisfies the equation \(\psi(\varphi(X))=X\) in either one of the following two cases:
- (a)
\(X\in\mathcal{C}_{\varphi}^{\mathrm{SOT}}(\mathcal{H})\) and either \(X_{i}=\sum_{k=0}^{\infty}\sum_{\alpha\in\mathbb{F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}[\varphi(X)]_{\alpha}\), \(i=1,\ldots,n\), where the convergence of the series is in the strong operator topology, or \(\varphi(X)\in\mathcal{C}_{\psi}^{\mathrm{rad}}(\mathcal {H})\) and
$$X_{i}=\text{SOT-}\lim_{r\to1}\sum _{k=0}^{\infty}\sum_{\alpha\in\mathbb {F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}r^{|\alpha|} \bigl[\varphi(X)\bigr]_{\alpha},\quad i=1,\ldots,n; $$ - (b)
\(X\in\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathcal{H})\) and either \(X_{i}=\sum_{k=0}^{\infty}\sum_{\alpha\in\mathbb{F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}[\widehat{\varphi}(X)]_{\alpha}\), \(i=1,\ldots,n\), where the convergence of the series is in the strong operator topology, or \(\widehat{\varphi}(X)\in\mathcal{C}_{\psi}^{\mathrm{rad}}(\mathcal{H})\) and
$$X_{i}=\text{SOT-}\lim_{r\to1}\sum _{k=0}^{\infty}\sum_{\alpha\in\mathbb {F}_{n}^{+},|\alpha|=k}c_{\alpha}^{(i)}r^{|\alpha|} \bigl[\widehat{\varphi }(X)\bigr]_{\alpha},\quad i=1,\ldots,n. $$
Definition 2.3
(see [28])
Let \(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\) be a positive regular free holomorphic function, and let \(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\) be an n-tuple of formal power series with model property. The noncommutative domain \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) is the set of all n-tuples of bounded linear operators \(X=(X_{1},\ldots,X_{n})\in B(\mathcal{H})^{n}\) such that \(\psi(\varphi(X))=X\) and
where the convergence is in the weak operator topology. Define the strict noncommutative domain
where the convergence is in the weak operator topology.
We define the noncommutative Hardy algebra \(H^{\infty}(\mathbb {D}_{f,\varphi})\) to be the WOT-closure of all noncommutative polynomials in \(M_{Z_{1}},\ldots,M_{Z_{n}}\) and the identity. Similarly, we can also define the noncommutative Hardy algebra \(R^{\infty}(\mathbb{D}_{f,\varphi})\) to be the WOT-closure of all noncommutative polynomials in \(R_{Z_{1}},\ldots,R_{Z_{n}}\) and the identity. Now we can define the strict noncommutative variety
where \(\mathcal{I}\) is a WOT-closed two-sided ideal of the noncommutative Hardy algebra \(H^{\infty}(\mathbb{D}_{f,\varphi})\).
2.3 Noncommutative Poisson kernel
If \(T=(T_{1},\ldots,T_{n})\in\mathbb{D}_{f,\varphi}(\mathcal{H})\), we define the positive linear mapping
where the convergence is in the weak operator topology. We say that \(T=(T_{1},\ldots,T_{n})\) is a pure n-tuple of operators in \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) if
The set of all pure elements of \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) is denoted by \(\mathbb{D}_{f,\varphi}^{\mathrm{pure}}(\mathcal{H})\). Notice that \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is in \(\mathbb{D}_{f,\varphi }^{\mathrm{pure}}(\mathbb{H}_{f}^{2}(\varphi))\). Moreover, we refer to the n-tuple \((M_{Z_{1}},\ldots, M_{Z_{n}})\) as the universal model associated with the abstract noncommutative domain \(\mathbb{D}_{f,\varphi}\). An n-tuple \(T\in\mathbb {D}_{f,\varphi}(\mathcal{H})\) is called completely non-coisometric (c.n.c.) if there is no vector \(h\in\mathcal{H}\), \(h\neq0\), such that
The set of all c.n.c. elements of \(\mathbb{D}_{f,\varphi}(\mathcal{H})\) is denoted by \(\mathbb{D}_{f,\varphi}^{\mathrm{cnc}}(\mathcal{H})\). Note that
Similarly, we have
Moreover, it is obvious that the n-tuple \((B_{1},\ldots,B_{n})\) is in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{pure}}(\mathcal{N}_{f,\varphi,\mathcal{I}})\), where \(B_{i}:=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\) for \(i=1,\ldots,n\). We refer to the n-tuple \((B_{1},\ldots,B_{n})\) as the universal model associated with the abstract noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}\).
We define the noncommutative Poisson kernel associated with the n-tuple \(T:=(T_{1},\ldots,T_{n})\in\mathbb{D}_{f,\varphi}(\mathcal{H})\) to be the operator \(K_{f,\varphi,T}:\mathcal{H}\to\mathbb{H}_{f}^{2}(\varphi)\otimes\overline {\Delta_{f,\varphi,T}(\mathcal{H})}\) defined by
where \(\Delta_{f,\varphi,T}:=(I-\varPhi_{f,\varphi,T}(I))^{\frac{1}{2}}\) and the coefficients \(b_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), are given by relation (2.1).
2.4 Characteristic function
We consider the full Fock space of \(H_{n}\) defined by
where \(H_{n}^{\otimes m}\) is the Hilbert tensor product of m copies of \(H_{n}\). Define the left creation operators \(S_{i}\), \(i = 1,\ldots,n\), acting on \(F^{2}(H_{n})\) by setting \(S_{i}\xi:= e_{i}\otimes\xi\), \(\xi\in F^{2}(H^{n})\). If \(A \in B(F^{2}(H_{n})\otimes\mathcal{G} ,F^{2}(H_{n})\otimes\mathcal{K} )\) and
then A is called multi-Toeplitz with respect to \(S_{1},\ldots,S_{n}\). Moreover, if \(A \in B(F^{2}(H_{n})\otimes\mathcal{G} , F^{2}(H_{n})\otimes\mathcal{K} )\) and
then A is called multi-analytic with respect to \(S_{1},\ldots,S_{n}\) (see [17, 19]). We remark that several results concerning the full Fock space \(F^{2}(H_{n})\) have been extended to the Hilbert space \(\mathbb{H}^{2}_{f}(\varphi)\) (see [25, 26, 28]). If \(A\in B(\mathbb{H}_{f}^{2}(\varphi)\otimes\mathcal{G}, \mathbb {H}_{f}^{2}(\varphi)\otimes\mathcal{K})\), and
then A is called multi-analytic with respect to \(M_{Z_{1}},\ldots ,M_{Z_{n}}\) (see Definition 3.1 of [28]). Indeed, this definition is an analogy.
Let \(f=\sum_{|\alpha|\ge1}a_{\alpha}X_{\alpha}\) be a positive regular free holomorphic function and define the set \(\varGamma:=\{\alpha\in\mathbb {F}_{n}^{+}:a_{\alpha}\neq0\}\) and \(N:=\operatorname{card}(\varGamma)\). If \(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\) is an n-tuple of formal power series with the model property and \(T:=(T_{1},\ldots,T_{n})\in \mathbb{D}_{f,\varphi}(\mathcal{H})\), we define the row operator
where the entries are arranged in the lexicographic order of \(\varGamma \subset\mathbb{F}_{n}^{+}\), and α̃ is the reverse of \(\alpha=g_{i_{1}}\cdots g_{i_{k}}\), i.e., \(\widetilde{\alpha}=g_{i_{k}}\cdots g_{i_{1}}\). Note that \(C_{f,\varphi,T}\) is an operator acting from \(\mathcal {H}^{(N)}\) (the completion of the direct sum of N copies of \(\mathcal {H}\)) to \(\mathcal{H}\).
Let \((M_{Z_{1}},\ldots,M_{Z_{n}})\) be the universal model associated with the abstract noncommutative domain \(\mathbb{D}_{f,\varphi}\). We introduce the characteristic function of an n-tuple \(T:=(T_{1},\ldots,T_{n})\in \mathbb{D}_{f,\varphi}(\mathcal{H})\) to be the multi-analytic operator with respect to \(M_{Z_{1}},\ldots,M_{Z_{n}}\),
with formal Fourier representation
where \(R_{\varphi_{1}},\ldots,R_{\varphi_{n}}\) are the right multiplication operators by the formal power series \(\varphi_{1},\ldots,\varphi_{n}\), respectively, on the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\). The defect operators associated with the row contraction \(C_{f,\varphi,T}\) are
and the defect spaces are \(\mathcal{D}_{C_{f,\varphi,T}}:=\overline {\Delta_{C_{f,\varphi,T}} \mathcal{H}}\) and \(\mathcal{D}_{C_{f,\varphi,T}^{*}}:= \overline{\Delta_{C_{f,\varphi,T}^{*}} \mathcal{H}^{(N)}}\).
3 Constrained characteristic functions
In this section, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions. Moreover, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements. Indeed, this result can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions.
Let \(T=(T_{1},\ldots,T_{n})\) be an n-tuple of operators in \(\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\). The constrained Poisson kernel is the operator \(K_{f,\varphi,T}^{(\mathcal{I})}:\mathcal{H}\to\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\) defined by
where \(K_{f,\varphi,T}\) is the noncommutative Poisson kernel associated with f, φ, and T.
First, we present some basic properties for the constrained Poisson kernel \(K_{f,\varphi,T}^{(\mathcal{I})}\) associated with f, φ, T, and \(\mathcal{I}\).
Theorem 3.1
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(T=(T_{1},\ldots,T_{n})\)is ann-tuple of operators in\(\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H})\), then the following statements hold:
- (i)
\(K_{f,\varphi,T}^{(\mathcal{I})}T_{i}^{*}=(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}})K_{f,\varphi,T}^{(\mathcal{I})}\), \(i=1,\ldots,n\);
- (ii)
\(K_{f,\varphi,T}^{(\mathcal{I})}\)is an isometry if and only ifTis pure,
where\(K_{f,\varphi,T}^{(\mathcal{I})}\)is the constrained Poisson kernel associated withf, φ, T, and\(\mathcal{I}\).
Proof
(i) According to the proof of Theorem 2.1 from [28], we know that
where \(K_{f,\varphi,T}\) is the noncommutative Poisson kernel associated with f, φ, and T. Hence, we have
for any polynomial p in \(M_{Z_{1}},\ldots,M_{Z_{n}}\). Assume that
is an element in the noncommutative Hardy algebra \(F^{\infty}(\mathcal {D}_{f})\). Then we deduce that
is in the noncommutative domain algebra \(\mathcal{A}(\mathcal{D}_{f})\). Moreover, since φ has model property, we have
where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the set \(\mathcal {C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\). Using (2.3), we conclude that
Therefore, we obtain
where the series is convergent in the operator norm topology. Hence, due to (3.1), we infer that
for any \(\phi(V_{1},\ldots,V_{n})\in F^{\infty}(\mathcal{D}_{f})\) and \(0< r<1\). Since \(T=(T_{1},\ldots,T_{n})\) is in \(\mathbb{D}_{f,\varphi}^{\mathrm{cnc}}(\mathcal {H})\) and \(M_{Z}=(M_{Z_{1}},\ldots,M_{Z_{n}})\) is in \(\mathbb{D}_{f,\varphi }^{\mathrm{pure}}(\mathbb{H}_{f}^{2}(\varphi))\), we deduce that \(\varphi(T)=(\varphi _{1}(T),\ldots,\varphi_{n}(T))\) is a completely non-coisometric n-tuple of operators in the noncommutative domain \(\mathcal{D}_{f}(\mathcal{H})\) and \(\varphi (M_{Z})=(\varphi_{1}(M_{Z}),\ldots,\varphi_{n}(M_{Z}))\) is a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathbb{H}_{f}^{2}(\varphi))\). Taking into account that
and using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus (see [24]), we infer that
for any \(\phi(V_{1},\ldots,V_{n})\in F^{\infty}(\mathcal{D}_{f})\). Using Proposition 4.2 from [28], we know that if \(\theta\in H^{\infty}(\mathbb{D}_{f,\varphi})\), there is \(\chi=\sum_{\alpha\in\mathbb {F}_{n}^{+}}c_{\alpha}V_{\alpha}\) in \(F^{\infty}(\mathcal{D}_{f})\) such that
Indeed, this implies that
Moreover, since \(T=(T_{1},\ldots,T_{n})\) is in \(\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\), we deduce that \(\varphi(T)=(\varphi_{1}(T), \ldots,\varphi_{n}(T))\) is also a completely non-coisometric n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{H})\). Using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus, we obtain that
This shows that
for any \(\omega\in H^{\infty}(\mathbb{D}_{f,\varphi})\). Consequently, we deduce that
for any \(\omega\in H^{\infty}(\mathbb{D}_{f,\varphi})\), \(h\in\mathcal {H}\), and \(d\in\mathcal{D}_{C_{f,\varphi,T}}\). Since \(\mathcal{I}\) is a WOT-closed two-sided ideal of \(H^{\infty}(\mathbb{D}_{f,\varphi})\), we have
Note that \(T\in\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal {H})\). Then we obtain
for any \(\omega\in\mathcal{I}\), \(h\in\mathcal{H}\), and \(d\in\mathcal {D}_{C_{f,\varphi,T}}\). Therefore, we conclude that
which implies that
On the other hand, since \(\mathcal{N}_{f,\varphi,\mathcal{I}}\) is an invariant subspace under \(M_{Z_{1}}^{*},\ldots,M_{Z_{n}}^{*}\), we have
According to Proposition 4.2 of [28], we know that, for any \(\nu\in H^{\infty}(\mathbb{D}_{f,\varphi})\), there exists \(\chi\in F^{\infty}(\mathcal{D}_{f})\) such that
Since \((B_{1},\ldots,B_{n})\) is in the noncommutative variety \(\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{pure}}(\mathcal{N}_{f,\varphi,\mathcal {I}})\), we obtain that \((\varphi_{1}(B),\ldots, \varphi_{n}(B))\) is a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{N}_{f,\varphi,\mathcal{I}})\). Consequently, using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus, we deduce that
for any \(\nu\in H^{\infty}(\mathbb{D}_{f,\varphi})\). Applying (3.2), (3.3), and (3.4), we infer that
for any \(\nu(B_{1},\ldots,B_{n})\in H^{\infty}(\mathcal{V}_{f,\varphi,\mathcal {I}})\). In particular, we have
(ii) Due to (3.3), we obtain
Hence, we deduce that
where \(\varPhi_{f,\varphi,T}^{\infty}(I):=\text{SOT-}\lim_{m\to\infty }\varPhi_{f,\varphi,T}^{m}(I)\). Therefore, (ii) holds. This completes the proof. □
We define the constrained characteristic function associated with an n-tuple \(T:=(T_{1},\ldots,T_{n})\in\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H})\) to be the multi-analytic operator with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\),
with the formal Fourier representation
where \(D_{i}=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}R_{\varphi _{i}}|_{\mathcal{N}_{f,\varphi,\mathcal{I}}}\), \(i=1,\ldots,n\), and \(R_{\varphi_{1}}, \ldots,R_{\varphi_{n}}\) are the right multiplication operators by the power series \(\varphi_{1},\ldots,\varphi_{n}\), respectively, on the Hilbert space \(\mathbb{H}_{f}^{2}(\varphi)\).
We provide a factorization result for the constrained characteristic function, which will play an important role in our investigation.
Theorem 3.2
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). Then
where\(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\)is the constrained characteristic function and\(K_{f,\varphi,T}^{(\mathcal{I})}\)is the corresponding constrained Poisson kernel.
Proof
Due to Theorem 6.1 of [28], we know that
According to the proof of Theorem 3.1, we have
Hence, we infer that
Since \(\mathcal{N}_{f,\varphi,\mathcal{I}}\) is an invariant subspace under \(R_{\varphi_{1}}^{*},\ldots,R_{\varphi_{n}}^{*}\), we obtain
and
Applying (3.6), (3.7), and (3.8), we deduce that
This completes the proof. □
If \(A\in B(\mathbb{H}_{f}^{2}(\varphi)\otimes\mathcal{G},\mathbb {H}_{f}^{2}(\varphi)\otimes\mathcal{K})\) is a multi-analytic operator and A is a partial isometry, then we call it inner multi-analytic.
In what follows, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})\) in terms of constrained characteristic functions.
Theorem 3.3
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(T:=(T_{1},\ldots,T_{n})\)is in the noncommutative variety\(\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H})\), then the following statements hold:
- (i)
Tis unitarily equivalent to then-tuple\(\widetilde {T}:=(\widetilde{T}_{1},\ldots,\widetilde{T}_{n})\in\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{cnc}}(\widetilde{\mathcal{H}})\)on the Hilbert space
$$\begin{aligned} \widetilde{\mathcal{H}}:={} & \bigl[(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\oplus\overline{\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}( \mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}})}\bigr] \\ & \ominus\bigl\{ \varTheta_{f,\varphi,T}^{(\mathcal{I})}x\oplus\Delta _{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}x:x\in\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}} \bigr\} , \end{aligned}$$where\(\Delta_{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}=(I-(\varTheta _{f,\varphi,T}^{(\mathcal{I})})^{*}\varTheta_{f,\varphi,T}^{(\mathcal {I})})^{\frac{1}{2}}\)and each operator\(\widetilde{T}_{i}\), \(i=1,\ldots,n\), is uniquely defined by the relation
$$\begin{aligned} & (P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}})\widetilde{T}_{i}^{*}z \\ &\quad= \bigl(B_{i}^{*}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}}\bigr) (P_{\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}})z,\quad z\in\widetilde{\mathcal{H}}, \end{aligned}$$where\(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}}|_{\widetilde{\mathcal{H}}}\)is an injective operator, \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}}\)is the orthogonal projection from the Hilbert space
$$\widetilde{\mathcal{K}}:=(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\oplus\overline{\Delta_{\varTheta _{f,\varphi,T}^{(\mathcal{I})}}( \mathcal{N}_{f,\varphi,\mathcal {I}}\otimes \mathcal{D}_{C_{f,\varphi,T}^{*}})} $$onto the subspace\(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}\), and\(B_{i}=P_{\mathcal{N}_{f,\varphi,\mathcal{I}}}M_{Z_{i}}|_{\mathcal {N}_{f,\varphi,\mathcal{I}}}\)for any\(i=1,\ldots,n\);
- (ii)
Tis in the noncommutative variety\(\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\)if and only if the constrained characteristic function\(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\)is an inner multi-analytic operator. In this case, Tis unitarily equivalent to then-tuple
$$\bigl(P_{\widetilde{\mathcal{H}}}(B_{1}\otimes I_{\mathcal{D}_{C_{f,\varphi ,T}}})|_{\widetilde{\mathcal{H}}}, \ldots,P_{\widetilde{\mathcal {H}}}(B_{n}\otimes I_{\mathcal{D}_{C_{f,\varphi,T}}})|_{\widetilde {\mathcal{H}}} \bigr), $$where\(P_{\widetilde{\mathcal{H}}}\)is the orthogonal projection from\(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}}\)onto the Hilbert space\(\widetilde{\mathcal{H}}:=(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{D}_{C_{f,\varphi,T}})\ominus\varTheta_{f,\varphi,T}^{(\mathcal {I})}(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}^{*}})\).
Proof
(i) We define the operator \(\varPsi:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\widetilde{\mathcal{K}}\) by setting
It is obvious that Ψ is an isometry and
Hence, we infer that
for any \(y\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal {D}_{C_{f,\varphi,T}}\), where \(P_{\widetilde{\mathcal{H}}}\) denotes the orthogonal projection from \(\widetilde{\mathcal{K}}\) onto \(\widetilde{\mathcal{H}}\). According to Theorem 3.2, we have
Therefore, using (3.10) and (3.11), we deduce that
On the other hand, due to (3.3), we obtain
Hence, if \(K_{f,\varphi,T}^{(\mathcal{I})}h=0\), then we have
Since T is in \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal {H})\), we infer that \(h=0\), which implies that \(K_{f,\varphi ,T}^{(\mathcal{I})}\) is an injective operator and range \((K_{f,\varphi,T}^{(\mathcal{I})})^{*}\) is dense in \(\mathcal{H}\).
Let \(z\in\widetilde{\mathcal{H}}\) and assume that \(z\perp P_{\widetilde {\mathcal{H}}}(y\oplus0)\) for any \(y\in\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\). Taking into account that
Consequently, we obtain \(z=0\). This shows that
Applying (3.12) and (3.13), we deduce that there exists a unique unitary operator \(W:\mathcal{H}\to\widetilde{\mathcal{H}}\) such that
Moreover, using (3.9) and Theorem 3.2 , we have
for any \(y\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}\). Since the range \((K_{f,\varphi,T}^{(\mathcal {I})})^{*}\) is dense in \(\mathcal{H}\), we infer that
Let \(\widetilde{T}_{i}:\widetilde{\mathcal{H}}\to\widetilde{\mathcal{H}}\) be the transform of \(T_{i}\) under the unitary operator \(W:\mathcal{H}\to \widetilde{\mathcal{H}}\), i.e.,
Since the constrained Poisson kernel \(K_{f,\varphi,T}^{(\mathcal{I})}\) is an injective operator, due to (3.14), we deduce that
is an injective operator acting from \(\widetilde{\mathcal{H}}\) to \(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\). Consequently, according to (3.14) and Theorem 3.1, we have
for any \(h\in\mathcal{H}\) and \(i=1,\ldots,n\). Hence, we obtain that
for any \(z\in\widetilde{\mathcal{H}}\) and \(i=1,\ldots,n\). Notice that \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi ,T}}}|_{\widetilde{\mathcal{H}}}\) is an injective operator. Then (3.15) uniquely determines each operator \(\widetilde{T}_{i}\), \(i=1,\ldots,n\).
(ii) First, assume that \(T=(T_{1},\ldots,T_{n})\in\mathcal{V}_{f,\varphi ,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\). Due to Theorem 3.1, we know that the constrained Poisson kernel \(K_{f,\varphi,T}^{(\mathcal{I})}:\mathcal{H}\to\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\) is an isometry. Hence, \(K_{f,\varphi,T}^{(\mathcal{I})}(K_{f,\varphi,T}^{(\mathcal {I})})^{*}\) is the orthogonal projection from \(\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}\) onto \(K_{f,\varphi,T}^{(\mathcal{I})}\mathcal{H}\). According to Theorem 3.2, we deduce that \(\varTheta_{f,\varphi ,T}^{(\mathcal{I})}(\varTheta_{f,\varphi,T}^{(\mathcal{I})})^{*}\) is also a projection, which implies that \(\varTheta_{f,\varphi ,T}^{(\mathcal{I})}\) is a partial isometry. This shows that \(\varTheta _{f,\varphi,T}^{(\mathcal{I})}\) is an inner multi-analytic operator.
Conversely, if \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) is an inner multi-analytic operator, then it is a partial isometry. Applying Theorem 3.2, we infer that \(K_{f,\varphi,T}^{(\mathcal{I})}\) is a partial isometry. Moreover, since T is in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\), due to (3.5), we deduce that \(K_{f,\varphi,T}^{(\mathcal{I})}\) is an injective operator, which implies that \(K_{f,\varphi,T}^{(\mathcal{I})}\) is an isometry. Therefore, using Theorem 3.1, we deduce that T is in \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{pure}}(\mathcal{H})\).
Now, we prove the last part of the theorem. Notice that \(u\oplus v\in \widetilde{\mathcal{K}}\) is in \(\widetilde{\mathcal{H}}\) if and only if
for any \(x\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}^{*}}\). Note that condition (3.16) is equivalent to
Since the operator \(\Delta_{\varTheta_{f,\varphi,T}^{(\mathcal{I})}}\) is the orthogonal projection from \(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}\) onto \([\operatorname{range} (\varTheta_{f,\varphi,T}^{(\mathcal {I})})^{*}]^{\perp}\), we have
Hence, (3.17) holds if and only if \((\varTheta_{f,\varphi,T}^{(\mathcal{I})})^{*}u=0\) and \(v=0\). Therefore, we conclude that
and
According to (3.15), we infer that
This completes the proof. □
Let \(\varPhi:\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {H}_{1}\to\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}_{2}\) and \(\varPhi':\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}'_{1}\to \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{H}'_{2}\) be two multi-analytic operators with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\), i.e.,
for any \(i=1,\ldots,n\). We say that Φ and \(\varPhi'\) coincide if there exist two unitary operators \(U_{j}\in B(\mathcal{H}_{j},\mathcal {H}_{j}')\), \(j=1,2\), such that
Applying Theorem 3.3, we can show that the constrained characteristic function \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) is a complete unitary invariant for the n-tuples of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\).
Theorem 3.4
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(T=(T_{1},\ldots,T_{n})\in\mathcal {V}_{f,\varphi,\mathcal{I}}^{\mathrm{cnc}}(\mathcal{H})\)and\(T'=(T'_{1},\ldots,T'_{n})\in\mathcal{V}_{f,\varphi,\mathcal {I}}^{\mathrm{cnc}}(\mathcal{H}')\), thenTand\(T'\)are unitarily equivalent if and only if their constrained characteristic functions\(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\)and\(\varTheta _{f,\varphi,T'}^{(\mathcal{I})}\)coincide.
Proof
First, we assume that \(\varTheta_{f,\varphi,T}^{(\mathcal{I})}\) and \(\varTheta_{f,\varphi,T'}^{(\mathcal{I})}\) coincide. Then there are two unitary operators \(U_{1}:\mathcal{D}_{C_{f,\varphi,T}}\to\mathcal{D}_{C_{f,\varphi,T'}}\) and \(U_{2}:\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\mathcal{D}_{C_{f,\varphi ,T'}^{*}}\) such that
Consequently, we have
and
Now we define the unitary operator \(W:\widetilde{\mathcal{K}}\to \widetilde{\mathcal{K}}'\) by setting
where \(\widetilde{\mathcal{K}}\) and \(\widetilde{\mathcal{K}}'\) were defined in Theorem 3.3. Notice that the operator \(\varPsi:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}^{*}}\to\widetilde{\mathcal{K}}\), defined by
and the corresponding \(\varPsi':\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T'}^{*}}\to\widetilde{\mathcal {K}}'\) satisfy the following relations:
and
where \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T}}}^{\widetilde{\mathcal{K}}}\) is the orthogonal projection from \(\widetilde{\mathcal{K}}\) onto \(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{D}_{C_{f,\varphi,T}}\). Hence, we have
which implies that \(W|_{\widetilde{\mathcal{H}}}:\widetilde{\mathcal {H}}\to\widetilde{\mathcal{H}}'\) is unitary. On the other hand, for any \(i=1,\ldots,n\),
Now, we assume that \(\widetilde{T}:=(\widetilde{T}_{1},\ldots,\widetilde {T}_{n})\) and \(\widetilde{T}':=(\widetilde{T}'_{1},\ldots,\widetilde {T}'_{n})\) are the model operators provided by Theorem 3.3 for T and \(T'\), respectively. Therefore, applying (3.18), (3.19), and (3.20), we deduce that
for any \(z\in\widetilde{\mathcal{H}}\) and \(i=1,\ldots,n\). Using the fact that \(P_{\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal {D}_{C_{f,\varphi,T'}}}^{\widetilde{\mathcal{K}}'}\) is an injective operator, we infer that
Due to Theorem 3.3, it is obvious that T and \(T'\) are unitarily equivalent.
Conversely, let \(\varOmega:\mathcal{H}\to\mathcal{H}'\) be a unitary operator such that
Note that \(T\in\mathcal{C}_{\varphi}^{\mathrm{SOT}}(\mathcal{H})\) or \(T\in\mathcal {C}_{\varphi}^{\mathrm{rad}}(\mathcal{H})\) and similar relations hold for \(T'\). Then we obtain
Now we define the unitary operator by setting
and
A simple calculation shows that
This completes the proof. □
4 Multivariable interpolation and invariant subspaces
In this section, we prove a Sarason-type commutant lifting theorem. As an application, we obtain the Nevanlinna–Pick-type interpolation result in our setting. Moreover, we provide a Beurling-type characterization of the joint invariant subspaces under the constrained weighted shifts \(B_{1},\ldots,B_{n}\).
For each \(i=1,\ldots,n\), we define the right multiplication operator \(R_{i}:\mathcal{F}_{f}^{2}\to\)\(\mathcal{F}_{f}^{2}\) by setting \(R_{i}\zeta=\zeta Z_{i}\), \(\zeta\in\mathcal{F}_{f}^{2}\). Using the results from [24], we know that \(R^{\infty}(\mathcal {D}_{f})\) is the WOT-closure of all polynomials in \(R_{1},\ldots,R_{n}\) and the identity. Moreover, we define the noncommutative Hardy algebra \(R^{\infty}(\mathbb {D}_{f,\varphi})\) to be the WOT-closure of all noncommutative polynomials in \(R_{Z_{1}},\ldots,R_{Z_{n}}\) and the identity.
The following result is a Sarason-type [32] commutant lifting theorem.
Theorem 4.1
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). For each\(j=1,2\), let\(\mathcal {K}_{j}\)be a Hilbert space, and let\(\mathcal{E}_{j}\subseteq\mathcal {N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}_{j}\)be an invariant subspace under\(B_{i}^{*}\otimes I_{\mathcal{K}_{j}}\), \(i=1,\ldots,n\). If\(X:\mathcal{E}_{1}\to\mathcal{E}_{2}\)is a bounded operator such that
then there exists
such that
Proof
First, note that the subspace \(\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}_{j}\) is invariant under \(M_{Z_{i}}^{*}\otimes I_{\mathcal{K}_{j}}\), and
Since \(\mathcal{E}_{j}\subseteq\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{K}_{j}\) is invariant under \(B_{1}^{*}\otimes I_{\mathcal {K}_{j}},\ldots,B_{n}^{*}\otimes I_{\mathcal{K}_{j}}\), it is also invariant under \(M_{Z_{1}}^{*}\otimes I_{\mathcal{K}_{j}},\ldots ,M_{Z_{n}}^{*}\otimes I_{\mathcal{K}_{j}}\), which implies that
Hence, we deduce that
According to Theorem 5.1 of [28], there exists a bounded operator \(\varPhi:\mathbb{H}^{2}_{f}(\varphi)\otimes\mathcal{K}_{1}\to \mathbb{H}^{2}_{f}(\varphi)\otimes\mathcal{K}_{2}\) with the property
and such that \(\varPhi^{*}\mathcal{E}_{2}\subseteq\mathcal{E}_{1}\), \(\varPhi ^{*}|_{\mathcal{E}_{2}}=X^{*}\), and \(\|\varPhi\|=\|X\|\). Since \(M_{\varphi_{i}}=\varphi_{i}(M_{Z_{1}},\ldots, M_{Z_{n}})\) for any \(i=1,\ldots,n\), we have
Notice that
Then we obtain
for any \(i=1,\ldots,n\). This shows that
for any \(i=1,\ldots,n\). Due to the discussion of Proposition 1.11 from [24], we infer that
Using Proposition 4.2 in [28], we know
Consequently, we infer that
Assume that \(\varPhi(R_{Z_{1}},\ldots,R_{Z_{n}}):=\varPhi\). This shows that we can find \(\varPhi(R_{Z_{1}},\ldots, R_{Z_{n}})\in R^{\infty}(\mathbb {D}_{f,\varphi})\mathrel{\overline{\otimes}}B(\mathcal{K}_{1},\mathcal{K}_{2})\) such that \(\varPhi(R_{Z_{1}},\ldots,R_{Z_{n}})^{*}\mathcal{E}_{2}\subseteq\mathcal{E}_{1}\),
Moreover, we assume that
Then we have \(\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}(\mathcal{V}_{f,\varphi ,\mathcal{I}})\mathrel{\overline{\otimes}}B(\mathcal{K}_{1},\mathcal{K}_{2})\). Notice that
and \(\mathcal{E}_{j}\subseteq\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}_{j}\). Using (4.2), we obtain
Applying again (4.2), we infer that
which shows that
This completes the proof. □
Applying Theorem 4.1, we can obtain the following Nevanlinna–Pick-type interpolation result in our setting.
Theorem 4.2
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). Let\(\lambda_{1},\ldots,\lambda_{k}\)bekdistinct points in\(\mathcal{V}_{f,\varphi,\mathcal{I}}^{<}(\mathbb {C})\), and let\(A_{1},\ldots,A_{k}\in B(\mathcal{K})\). Then there exists\(\varPhi(C_{1},\ldots,C_{n})\in R^{\infty}(\mathcal{V}_{f,\varphi,\mathcal {I}})\mathrel{\overline{\otimes}}B(\mathcal{K})\)such that
if and only if the operator matrix
is positive semidefinite, where
Proof
Let \(\lambda_{j}:=(\lambda_{j_{1}},\ldots,\lambda_{j_{n}})\), \(j=1,\ldots,k\), be k distinct points in \(\mathcal{V}_{f,\varphi,\mathcal{I}}^{<}(\mathbb {C})\), and let
where the coefficients \(b_{\alpha}\), \(\alpha\in\mathbb{F}_{n}^{+}\), are given by relation (2.1). Since φ has model property, we have
where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the set \(\mathcal {C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\). Due to Proposition 4.2 of [28], for any \(\omega\in\mathcal {I}\subseteq H^{\infty}(\mathbb{D}_{f,\varphi})\), there exists \(\chi=\sum_{\alpha\in\mathbb{F}_{n}^{+}}c_{\alpha}V_{\alpha}\in F^{\infty}(\mathcal {D}_{f})\) such that
Using (4.4) and (4.5), we infer that
Since \(\mathcal{I}\) is a WOT-closed two-sided ideal of \(H^{\infty}(\mathbb {D}_{f,\varphi})\), we obtain
This shows that
According to Theorem 4.4 of [28], we have
Moreover, notice that
Hence, we deduce that the subspace
is invariant under \(B_{i}^{*}\) for any \(i=1,\ldots,n\), and \(\mathcal {M}\subseteq\mathcal{N}_{f,\varphi,\mathcal{I}}\). Now, we define the operators \(X_{i}\in B(\mathcal{M}\otimes\mathcal{K})\) by setting
Note that \(Z_{f,\varphi}^{(\lambda_{1})},\ldots,Z_{f,\varphi}^{(\lambda _{k})}\) are linearly independent. Then we can define an operator \(T\in B(\mathcal{M}\otimes\mathcal{K})\) by setting
for any \(h\in\mathcal{K}\) and \(j=1,\ldots,k\). A simple calculation shows that
Taking into account that \(\mathcal{M}\otimes\mathcal{K}\) is a co-invariant subspace under \(B_{i}\otimes I_{\mathcal{K}}\), \(i=1,\ldots,n\). Due to Theorem 4.1, we can find \(\varPhi(R_{Z_{1}},\ldots ,R_{Z_{n}})\in R^{\infty}(\mathbb{D}_{f,\varphi})\mathrel{\overline{\otimes}}B(\mathcal{K})\) such that
has the properties
and
In what follows, we prove
where \(Z_{f,\varphi}^{(\lambda)}\) is given by relation (4.4). Indeed, a straightforward computation reveals that
Consequently, we obtain
for any \(i=1,\ldots,n\). Moreover, due to the proof of Theorem 2.1 from [28], we have
for any \(i=1,\ldots,n\). Hence, we conclude that
for any \(i=1,\ldots,n\). Since \(\lambda\in\mathbb{D}_{f,\varphi }^{<}(\mathbb{C})\), we obtain \(\lambda_{i}=\psi_{i}(\varphi(\lambda))\) for any \(i=1, \ldots,n\). Therefore, we infer that
for any \(i=1,\ldots,n\). This proves our assertion. Since \(\lambda _{1},\ldots,\lambda_{k}\) are k distinct points in \(\mathcal{V}_{f,\varphi ,\mathcal{I}}^{<}(\mathbb{C})\subseteq\mathbb{D}_{f,\varphi}^{<}(\mathbb{C})\), we have \(R_{Z_{i}}^{*}Z_{f,\varphi}^{(\lambda_{j})}=\overline{\lambda _{ji}}Z_{f,\varphi}^{(\lambda_{j})}\), \(i=1,\ldots,n\); \(j=1, \ldots,k\). This shows that
for any \(\nu(R_{Z_{1}},\ldots,R_{Z_{n}})\in R^{\infty}(\mathbb{D}_{f,\varphi })\). Hence, we deduce that
Using (4.6), we obtain
for any \(x,y\in\mathcal{K}\) and \(j=1,\ldots,k\). Moreover, notice that
for any \(x,y\in\mathcal{K}\) and \(j=1,\ldots,k\). Since \(\varphi(\lambda _{1}),\ldots,\varphi(\lambda_{k})\) are in the strict noncommutative domain \(\mathcal{D}_{f,<}(\mathbb{C})\), we infer that
for any \(i,j=1,\ldots,k\). Hence, applying (4.7), (4.8), and (4.9), we conclude that \(\varPhi(\lambda_{j})=A_{j}\), \(j=1,\ldots,k\), if and only if \(\varPhi(C_{1},\ldots,C_{n})^{*}|_{\mathcal{M}\otimes\mathcal {K}}=T^{*}\).
Since \(\|\varPhi(C_{1},\ldots,C_{n})\|=\|T\|\), it is clear that
On the other hand, for any \(h_{1},\ldots,h_{k}\in\mathcal{K}\), we have
Consequently, we deduce that \(\|\varPhi(C_{1},\ldots,C_{n})\|\le1\) if and only if matrix (4.3) is positive semidefinite. This completes the proof. □
The following result is a noncommutative multivariable version of a result of Rosenblum and Rovnyak [31].
Theorem 4.3
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). If\(X\in B(\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{K})\)is a self-adjoint operator, then the following statements are equivalent:
- (i)
\(\varPhi_{f,\varphi,B\otimes I_{\mathcal{K}}}(X)\le X\), where\(B\otimes I_{\mathcal{K}}:=(B_{1}\otimes I_{\mathcal{K}},\ldots,B_{n}\otimes I_{\mathcal{K}})\);
- (ii)
there are a Hilbert space\(\mathcal{G}\)and a multi-analytic operator\(\varPhi:\mathcal{N}_{f,\varphi,\mathcal {I}}\otimes\mathcal{G}\to\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes \mathcal{K}\)with respect to the constrained weighted shifts\(B_{1},\ldots,B_{n}\)such that\(X=\varPhi\varPhi^{*}\).
Proof
First, we prove that (i) ⇒ (ii). Since \((B_{1},\ldots,B_{n})\) is a pure n-tuple of operators in the noncommutative variety \(\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal {N}_{f,\varphi,\mathcal{I}})\) and
we deduce that
Notice that
Then we obtain \(X\ge0\). Let \(\mathcal{M}:=\overline{\operatorname{range} X^{\frac{1}{2}}}\) and define
for any \(i=1,\ldots,n\). Note that
for any \(\xi\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}\). Hence, we obtain that
for any \(i=1,\ldots,n\). Since f is a positive regular free holomorphic function, each operator \(Q_{i}\), \(i=1,\ldots,n\), can be uniquely extended to a bounded operator (also denoted by \(Q_{i}\)) on \(\mathcal{M}\). Denoting \(A_{i}:=Q_{i}^{*}\) for any \(i=1,\ldots,n\), we have
where the convergence is in the weak operator topology. Setting \(\phi _{A}(X):=\sum_{|\alpha|\ge1}a_{\alpha}A_{\alpha}X A_{\alpha}^{*}\) (the convergence is in the weak operator topology) and using (4.10), we infer that
for any \(\xi\in\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}\), which implies that
This shows that \(A:=(A_{1},\ldots,A_{n})\) is a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{M})\). According to Proposition 4.2 of [28], we know that \(\mathcal{I}\) is a WOT-closed two-sided ideal of \(H^{\infty}(\mathbb{D}_{f,\varphi})\) if and only if there is a WOT-closed two-sided ideal J of \(F^{\infty}(\mathcal{D}_{f})\) such that
Taking into account that
Then, for any \(\chi\in J\), we obtain
for any \(r\in(0,1)\). Moreover, since \((A_{1},\ldots,A_{n})\) is a pure n-tuple of operators in the noncommutative domain \(\mathcal{D}_{f}(\mathcal{M})\) and \((\varphi _{1}(B),\ldots,\varphi_{n}(B))\) is also a pure n-tuple of operators in \(\mathcal{D}_{f}(\mathcal{N}_{f,\varphi,\mathcal{I}})\), using \(F^{\infty}(\mathcal{D}_{f})\)-functional calculus (see [24]), we have
for any \(\chi\in J\). Since \(X^{\frac{1}{2}}\) is an injective operator on \(\mathcal{M}\), we infer that
Consequently, we deduce that \((A_{1},\ldots,A_{n})\) is a pure n-tuple of operators in the noncommutative variety \(\mathcal{V}_{f,J}(\mathcal {M})\), where
Applying the appropriate result from [24], we know that the noncommutative Poisson kernel \(K_{f,A}:\mathcal{M}\to\mathbb {H}_{f}^{2}(\varphi)\otimes\mathcal{G}\) (\(\mathcal{G}\) is an appropriate Hilbert space) defined by
where \(\Delta_{f,A}:=(I-\sum_{|\alpha|\ge1}a_{\alpha}A_{\alpha}A_{\alpha}^{*})^{\frac{1}{2}}\) is an isometry with the properties that
for any \(i=1,\ldots,n\). Now we define
where the constrained Poisson kernel \(K_{f,A,\mathcal{I}}:\mathcal{M}\to \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}\) is defined by
Since φ has the model property, we have
where \((M_{Z_{1}},\ldots,M_{Z_{n}})\) is either in the set \(\mathcal {C}_{\varphi}^{\mathrm{SOT}}(\mathbb{H}_{f}^{2}(\varphi))\) or \(\mathcal{C}_{\varphi}^{\mathrm{rad}}(\mathbb{H}_{f}^{2}(\varphi))\). Hence, we obtain
Therefore, using (4.11) and (4.12), we infer that
for any \(i=1,\ldots,n\). On the other hand, notice that
for any \(i=1,\ldots,n\). Then we conclude that each operator \(B_{i}\), \(i=1,\ldots,n\), is in the SOT-closure of all polynomials in \(\varphi _{1}(B),\ldots, \varphi_{n}(B)\) and the identity. Consequently, we obtain that
This shows that Φ is a multi-analytic operator with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\). Moreover, since the constrained Poisson kernel \(K_{f,A,\mathcal{I}}\) is an isometry, we deduce that
Now, we prove that (ii) ⇒ (i). Note that \((B_{1},\ldots ,B_{n})\in\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{N}_{f,\varphi ,\mathcal{I}})\). Then we have
where the convergence is in the weak operator topology. This completes the proof. □
As an application, we obtain a Beurling-type characterization of the invariant subspaces under the constrained weighted shifts \(B_{1},\ldots,B_{n}\).
Theorem 4.4
Let\(f:=\sum_{\alpha\in\mathbb{F}_{n}^{+}}a_{\alpha}Z_{\alpha}\)be a positive regular free holomorphic function, and let\(\varphi=(\varphi_{1},\ldots ,\varphi_{n})\)be ann-tuple of formal power series with model property. Let\(\mathcal{I} \neq H^{\infty}(\mathbb{D}_{f,\varphi})\)be a WOT-closed two-sided ideal of the noncommutative Hardy algebra\(H^{\infty}(\mathbb{D}_{f,\varphi})\). A subspace\(\mathcal{M}\subseteq \mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{K}\)is invariant under\(B_{i}\otimes I_{\mathcal{K}}\), \(i=1,\ldots,n\), if and only if there are a Hilbert space\(\mathcal{G}\)and an inner multi-analytic operator
with respect to the constrained weighted shifts\(B_{1},\ldots,B_{n}\)such that
Proof
First, we assume that \(\mathcal{M}\subseteq\mathcal{N}_{f,\varphi ,\mathcal{I}}\otimes\mathcal{K}\) is invariant under \(B_{1}\otimes I_{\mathcal{K}},\ldots,B_{n}\otimes I_{\mathcal{K}}\). Notice that
and \((B_{1},\ldots,B_{n})\in\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal {N}_{f,\varphi,\mathcal{I}})\). Then we have
According to Theorem 4.3, there are a Hilbert space \(\mathcal{G}\) and a multi-analytic operator
with respect to the constrained weighted shifts \(B_{1},\ldots,B_{n}\) such that \(P_{\mathcal{M}}=\varPhi\varPhi^{*}\). Moreover, since \(P_{\mathcal{M}}\) is an orthogonal projection, we deduce that Φ is a partial isometry and \(\mathcal{M}=\varPhi[\mathcal{N}_{f,\varphi,\mathcal{I}}\otimes\mathcal{G}]\). The converse is obvious. This completes the proof. □
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Hu, J., Wang, M. & Wang, W. Constrained characteristic functions, multivariable interpolation, and invariant subspaces. J Inequal Appl 2020, 146 (2020). https://doi.org/10.1186/s13660-020-02412-x
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DOI: https://doi.org/10.1186/s13660-020-02412-x