A characteristic function is a special operator-valued analytic function defined on the open unit ball of $\mathbb{C}^n$ associated with an $n$-tuple of commuting row contraction on some Hilbert space. In this paper, we continue our study of the representations of $n$-tuples of commuting row contractions on Hilbert spaces, which have polynomial characteristic functions. Gleason's problem plays an important role in the representations of row contractions. We further complement the representations of our row contractions by proving theorems concerning factorizations of characteristic functions. We also emphasize the importance and the role of the noncommutative operator theory and noncommutative varieties to the classification problem of polynomial characteristic functions.
This paper extends the trace formulas of [5] with perturbations in normed ideals of $B(H)$ to multivariate functions of commuting contractions admitting a dilation to commuting normal contractions.
In this work, we give a description of Taylor spectrum of commuting 2- contractions in terms of characteristic functions of such contractions.The case of a single obtained by B.Sz.Nagy and C. Foias is generalised in this work.
It is well-known that an $n$-tuple $(n\ge 3)$ of commuting contractions does not posses an isometric dilation, in general. Considering a class of $n$-tuple of commuting contractions satisfying certain positivity assumption, we construct their isometric dilations and consequently establish their von Neumann inequality. The positivity assumption is related to Brehmer positivity and motivated by the study of isometric dilations of operator tuples in [4].
Given a bounded operator $Q$ on a Hilbert space $\mathcal{H}$, a pair of bounded operators $(T_1, T_2)$ on $\mathcal{H}$ is said to be $Q$-commuting if one of the following holds: \[ T_1T_2=QT_2T_1 \text{ or }T_1T_2=T_2QT_1 \text{ or }T_1T_2=T_2T_1Q. \] We give an explicit construction of isometric dilations for pairs of $Q$-commuting contractions for unitary $Q$, which generalizes the isometric dilation of Ando [2] for pairs of commuting contractions. In particular, for $Q=qI_{\mathcal{H}}$, where $q$ is a complex number of modulus $1$, this gives, as a corollary, an explicit construction of …
Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model theory and generalizations to commuting tuples of $C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivaria…
One of the most important results in operator theory is Andô's \cite{ando} generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Schäffer \cite{sfr} and Douglas \cite{Doug-Dilation}) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an Andô dilation for a commuting pair $(T_1,T_2)$ of contractions, except in some special cases \cite{A-M-Dist-Var, D-S, D-S-S}. In this paper, we give two new proofs of Andô's dilation theorem by giving…
The collection $\frk{C}_3$ of all triples of commuting contractions forms a family in the sense of Agler, and so has an "optimal" model $\pd\frk{C}_3$ generated by its extremal elements. A given $T\in\frk{C}_3$ is extremal if every $X\in\frk{C}_3$ extending $T$ is an extension by direct sum. We show that many of the known examples of triples in $\frk{C}_3$ that fail to have coisometric extensions are in fact extremal.
We develop a $d$-variable analog of the two-component de Bran-ges-Rovnyak reproducing kernel Hilbert space associated with a Schur-class function on the unit disk. In this generalization, the unit disk is replaced by the unit ball in $d$-dimensional complex Euclidean space, and the Schur class becomes the class of contractive multipliers on the Drury-Arveson space over the ball. We also develop some results on a model theory for commutative row contractions which are not necessarily completely noncoisometric (the case considered in earlier work of Bhattacharyya, Eschmeier and Sarkar)
If $T= \big[ T_1 ... T_n\big]$ is a row contraction with commuting entries, and the Arveson dilation is $\tilde T= \big[ \tilde T_1 ... \tilde T_n\big]$, then any operator $X$ commuting with each $T_i$ dilates to an operator $Z$ of the same norm which commutes with each $\tilde T_i$.
In the deBranges-Rovnyak functional model for contractions on Hilbert space, any completely non-coisometric (CNC) contraction is represented as the adjoint of the restriction of the backward shift to a deBranges-Rovnyak space, $\mathscr{H} (b)$, associated to a contractive analytic operator-valued function, $b$, on the open unit disk. We extend this model to a large class of CNC row contractions of several copies of a Hilbert space into itself (including all CNC row contractions with commuting component operators). Namely, we completely characterize the set of all CNC row contractions, $T$, wh…
The goal of the paper is to study the structure of the k-tuples of doubly $Λ$-commuting row isometries and the $C^*$-algebras they generate from the point of view of noncommutative multivariable operator theory. We obtain Wold decompositions, in this setting, and use them to classify the $k$-tuples of doubly $Λ$-commuting row isometries up to a unitary equivalence. We introduce a universal model in this setting, describe its invariant subspaces, and develop a dilation theory on $Λ$-polyballs.
Absolutely continuous commuting row contractions admit a weak-$*$ continuous functional calculus. Building on recent work describing the first and second dual spaces of the closure of the polynomial multipliers on the Drury-Arveson space, we give a complete characterization of these commuting row contractions in measure theoretic terms. We also establish that completely non-unitary row contractions are necessarily absolutely continuous, in direct parallel with the case of a single contraction. Finally, we consider refinements of this question for row contractions that are annihilated by a give…
The symmetrization map $π:\mathbb C^2\rightarrow \mathbb C^2$ is defined by $ π(z_1,z_2)=(z_1+z_2,z_1z_2). $ The closed symmetrized bidisc $Γ$ is the symmetrization of the closed unit bidisc $\overline{\mathbb D^2}$, that is, \[ Γ= π(\overline{\mathbb D^2})=\{ (z_1+z_2,z_1z_2)\,:\, |z_i|\leq 1, i=1,2 \}. \] A pair of commuting Hilbert space operators $(S,P)$ for which $Γ$ is a spectral set is called a $Γ$-contraction. Unlike the scalars in $Γ$, a $Γ$-contraction may not arise as a symmetrization of a pair of commuting contractions, even not as a symmetrization of a pair of commuting bounded op…
This paper is devoted to prove the S. L. Singh's common fixed point Theorem for commuting mappings in cone metric spaces. In this framework, we introduce the notions of Generalized Kannan Con- traction, Generalized Zamfirescu Contraction and Generalized Weak Contraction for a pair of mappings, proving afterward their respective fixed point results.
A particular case of the fundamental Sz.-Nagy--Foias functional model for a contraction states that a pure contraction always dilates to a pure isometry. We are interested in the similar question for pairs, more precisely: does a pair of commuting pure contractions always dilate to a pair of commuting pure isometries? The purpose of this article is to identify pairs of commuting pure contractions for which the above question has an affirmative answer. Our method is based on an explicit structure of isometric dilation for pure pairs of commuting contractions obtained in a recent work by Das, Sa…
There are three main results in this paper. First, we find an easily computable and simple condition which is necessary and sufficient for a commuting tuple of contractions to possess a non-zero Toeplitz operator. This condition is just that the adjoint of the product of the contractions is not pure. On one hand this brings out the importance of the product of the contractions and on the other hand, the non-pureness turns out to be equivalent to the existence of a pseudo-extension to a tuple of commuting unitaries. The second main result is a commutant pseudo-extension theorem obtained by stud…
The purpose of the paper is to obtain estimates for differences of functions of two pairs of commuting contractions on Hilbert space. In particular, Lipschitz type estimates, Hölder type estimates, Schatten--von Neumann estimates are obtained. The results generalize earlier known results for functions of self-adjoint operators, normal operators, contractions and dissipative operators.
An example due to Pisier shows that two commuting, completely polynomially bounded Hilbert space operators may not be simultaneously similar to contractions. Thus, while each operator is individually similar to a contraction, the pair is not jointly similar to a pair of commuting contractions. We show that this phenomenon does not occur in finite dimensions. More precisely, we show that a finite family of power bounded commuting matrices is always jointly similar to a family of contractions. In fact, the result can be extended to infinite families satisfying certain uniformity conditions. Our …
It is known that pure row contractions with one-dimensional defect spaces can be classified up to unitary equivalence by compressions of the standard $d$-shift acting on the full Fock space. Upon settling for a softer relation than unitary equivalence, we relax the defect condition and simply require the row contraction to admit a cyclic vector. We show that cyclic pure row contractions can be "transformed" (in a precise technical sense) into compressions of the standard $d$-shift. Cyclic decompositions of the underlying Hilbert spaces are the natural tool to extend this fact to higher multipl…
The goal of the present paper is to push Sz.-Nagy--Foias model theory for a completely nonunitary Hilbert-space contraction operator $T$, to the case of a commuting pair of contraction operators $(T_1, T_2)$ having product $T = T_1 T_2$ which is completely nonunitary. The idea is to use the Sz.-Nagy-Foias functional model for $T$ as the model space also for the commutative tuple ($T_1, T_2)$ with $T = T_1 T_2$ equal to the usual Sz.-Nagy--Foias model operator, and identify what added structure is required to classify such commutative contractive factorizations $T = T_1 T_2$ up to unitary equiv…
In this paper, we introduce a unitary invariant $Γ$ defined on the unit ball of $B(H)^n$ in terms of the characteristic function, the noncommutative Poisson kernel, and the defect operator associated with a row contraction. We show that $Γ$ detects the pure row isometries and completely classify them up to a unitary equivalence. We also show that $Γ$ detects the pure row contractions with polynomial characteristic functions and completely non-coisometric row contractions. In particular, we show that any completely non-coisometric row contraction with constant characteristic function is homogen…
We develop a Sz.-Nagy--Foias-type functional model for a commutative contractive operator tuple $\underline{T} = (T_1, \dots, T_d)$ having $T = T_1 \cdots T_d$ equal to a completely nonunitary contraction. We identify additional invariants ${\mathbb G}_\sharp, {\mathbb W}_\sharp$ in addition to the Sz.-Nagy--Foias characteristic function $Θ_T$ for the product operator $T$ so that the combined triple $({\mathbb G}_\sharp, {\mathbb W}_\sharp, Θ_T)$ becomes a complete unitary invariant for the original operator tuple $\underline{T}$. For the case $d \ge 3$ in general there is no commutative isome…
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 condtions for a Wold decomposition based on the Lebesgue decomposition of the row-isometries.
Motivated by Ball, Li, Timotin and Trent's Schur-Agler class version of commutant lifting theorem, we introduce a class, denoted by $\mathcal{P}_n(\mathcal{H})$, of $n$-tuples of commuting contractions on a Hilbert space $\mathcal{H}$. We always assume that $n \geq 3$. The importance of this class of $n$-tuples stems from the fact that the von Neumann inequality or the existence of isometric dilation does not hold in general for $n$-tuples, $n \geq 3$, of commuting contractions on Hilbert spaces (even in the level of finite dimensional Hilbert spaces). Under some rank-finiteness assumptions, w…
For a contraction P and a bounded commutant S of P, we seek a solution X of the operator equation S-S*P = (I-P*P)^1/2 X(I-P*P) 1/2, where X is a bounded operator on Ran(I-P*P) 1/2 with numerical radius of X being not greater than 1. A pair of bounded operators (S,P) which has the domain \Gamme = {(z 1 +z 2, z 1z 2) : |z1|{\leq} 1, |z2| {\leq}1} {\subseteq} C2 as a spectral set, is called a \Gamme-contraction in the literature. We show the existence and uniqueness of solution to the operator equation above for a Γ-contraction (S,P). This allows us to construct an explicit Γ-isometric dilation o…
We establish the existence and uniqueness of finite free resolutions - and their attendant Betti numbers - for graded commuting d-tuples of Hilbert space operators. Our approach is based on the notion of free cover of a (perhaps noncommutative) row contraction. Free covers provide a flexible replacement for minimal dilations that is better suited for higher-dimensional operator theory. For example, every graded d-contraction that is finitely multi-cyclic has a unique free cover of finite type - whose kernel is a Hilbert module inheriting the same properties. This contrasts sharply with what ca…
We define four 3x3 commuting contractions which do not dilate to commuting isometries. However they do satisfy the scalar von Neumann inequality. These matrices are all nilpotent of order 2. We also show that any three $3\times3$ commuting contractions which are scalar plus nilpotent of order 2 do dilate to commuting isometries.
We develop a dilation theory for row contractions subject to constraints determined by sets of noncommutative polynomials. Under natural conditions on the constraints, we have uniqueness for the minimal dilation. A characteristic function is associated with any (constrained) row contraction. For pure constrained row contraction, we show that the characteristic function is a complete unitary invariant and provide a model. We also show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction can be express…
Consider a nonzero contraction $T$ and a bounded operator $X$ satisfying $TX=qXT$ for a complex number $q$. There are some interesting results in the literature on $q$-commuting dilation and $q$-commutant lifting of such pair $(T,X)$ when $|q|=1$. Here we improve a few of them to the class of scalars $q$ satisfying $|q|\leq \dfrac{1}{\|T\|}$.
We study similarity classes of commuting row contractions annihilated by what we call higher order vanishing ideals of interpolating sequences. Our main result exhibits a Jordan-type direct sum decomposition for these row contractions. We illustrate how the family of ideals to which our theorem applies is very rich, especially in several variables. We also give two applications of the main result. First, we obtain a purely operator theoretic characterization of interpolating sequences. Second, we classify certain classes of cyclic commuting row contractions up to quasi-similarity in terms of t…
Let $A_1,\ldots,A_d$ be a $d$-tuple of commuting dissipative operators on a separable Hilbert space $\mathcal{H}$. Using the theory of operator vessels and their associated systems, we give a construction of a dilation of the multi-parameter semigroup of contractions on $\mathcal{H}$ given by $e^{i \sum_{j=1}^d t_j A_j}$.
The fundamental result of B. Sz. Nazy states that every contraction has a coisometric extension and a unitary dilation. The isometric dilation of a contraction on a Hilbert space motivated whether this theory can be extended sensibly to families of operators. It is natural to ask whether this idea can be generalized, where the contraction $T$ is substituted by a commuting $n$-tuples of operators $(S_1,\cdots, S_n)$ acting on some Hilbert space having $Γ_n$ as a spectral set. We derive the necessary conditions for the existence of a $Γ_n$-isometric dilation for $Γ_n$-contractions. Also we discu…
Recent results have shown that any closed operator $A$ commuting with the backwards shift $S^*$ restricted to $K ^2_u := H^2 \ominus u H^2$, where $u$ is an inner function, can be realized as a Nevanlinna function of $S^*_u := S^* |_{K^2_u}$, $A = φ(S^*_u)$, where $φ$ belongs to a certain class of Nevanlinna functions which depend on $u$. In this paper this result is generalized to show that given any contraction $T$ of class $C_0 (N)$, that any closed (and not necessarily bounded) operator $A$ commuting with the commutant of $T$ is equal to $φ(T)$ where $φ$ belongs to a certain class of Nevan…
We give an entire free holomorphic function $f$ which is unbounded on the row ball. That is, we give a holomorphic free noncommutative function which is continuous in the free topology developed by Agler and McCarthy but is unbounded on the set of row contractions.
We obtain dilation results which simultaneously generalize Sz.-Nagy dilation theorem for contractions, Ando's dilation theorem for commuting contractions, Sz.-Nagy--Foias commutant lifting theorem, and Schur's representation for the unit ball of H^\infty, in the framework of noncommutative varieties in several variables and Poisson kernels on Fock spaces. This leads to inequalities which, in the particular case of two contractions, are sharper than Ando's inequality and Agler-McCarthy's inequality in the setting of commuting contractive matrices or arbitrary commuting contractions of class C_0…
Ando's theorem states that any pair of commuting contractions on a Hilbert space can be dilated to a pair of commuting unitaries. Parrott presented an example showing that an analogous result does not hold for a triple of pairwise commuting contractions. We generalize both of these results as follows. Any n-tuple of contractions that commute according to a graph without a cycle can be dilated to an n-tuple of unitaries that commute according to that graph. Conversely, if the graph contains a cycle, we construct a counterexample.
Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold Decomposition for partial isometries to describe the models for these dilations, and discuss how the basic properties of a dilation depend…
We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. Our characteristic function is a complete unitary invariant for such tuples. Thereafter we extend the analysis of characteristic functions to contractive liftings of row contractions. We apply our theory to completely positive maps and explore applications t…
We show that the constrained characteristic function is a complete unitary invariant for J-constrained completely non-coisometric (c.n.c.) row contractions, where J is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra. We provide a model for this class of row contractions in terms of the constrained characteristic functions. In particular, we obtain a model theory for q-commuting c.n.c. row contractions.
An n-tuple (n \geq 2), T = (T_1, \ldots, T_n), of commuting bounded linear operators on a Hilbert space \mathcal{H} is doubly commuting if T_i T_j^* = T_j^* T_i for all $1 \leq i < j \leq n$. If in addition, each T_i \in C_{\cdot 0}, then we say that T is a doubly commuting pure tuple. In this paper we prove that a doubly commuting pure tuple $T$ can be dilated to a tuple of shift operators on some suitable vector-valued Hardy space H^2_{\mathcal{D}_{T^*}}(\mathbb{D}^n). As a consequence of the dilation theorem, we prove that there exists a closed subspace \mathcal{S}_T of the form \[\mathcal{…
A commuting tuple of $n$ operators $(S_1, \dots, S_{n-1}, P)$ defined on a Hilbert space $\mathcal{H}$, for which the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n}z_iz_j, \dots, \prod_{i=1}^{n}z_i \right) : |z_i|\leq 1, i=1, \dots, n \right\} \] is a spectral set is called a $Γ_n$-contraction. Also a triple of commuting operators $(A,B,P)$ for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called an $\mathbb E$-contraction, where \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3 \neq 0 \quad \fora…
Based on a Wold decomposition for families of partial isometries and projections of Cuntz-Krieger-Toeplitz-type, we extend several fundamental theorems from the case of single vertex graphs to the general case of countable directed graphs with no sinks. We prove a Szego-type factorization theorem for CKT families, which leads to information on the structure of the unit ball in free semigroupoid algebras, and show that joint similarity implies joint unitary equivalence for such families. For each graph we prove a generalization of von Neumann's inequality which applies to row contractions of op…
A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $Γ_n$ is a spectral set is called a $Γ_n$-contraction. We show that every $Γ_n$-contraction admits a decomposition into a $Γ_n$-unitary and a completely non-unitary $Γ_n$-contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $Γ_n$ and $Γ_n$-contractions.
We show that every multivariable contractive weighted shift dilates to a tuple of commuting unitaries, and hence satisfies von Neumann's inequality. This answers a question of Lubin and Shields. We also exhibit a closely related $3$-tuple of commuting contractions, similar to Parrott's example, which does not dilate to a $3$-tuple of commuting unitaries.
We present some general theorems about operator algebras that are algebras of functions on sets, including theories of local algebras, residually finite dimensional operator algebras and algebras that can be represented as the scalar multipliers of a vector-valued reproducing kernel Hilbert space. We use these to further develop a quantized function theory for various domains that extends and unifies Agler's theory of commuting contractions and the Arveson-Drury-Popescu theory of commuting row contractions. We obtain analogous factorization theorems, prove that the algebras that we obtain are …
The celebrated Sz.-Nagy and Foias and Ando theorems state that a single contraction, or a pair of commuting contractions, acting on a Hilbert space always possesses isometric dilation and subsequently satisfies the von Neumann inequality for polynomials in $\mathbb{C}[z]$ or $\mathbb{C}[z_1, z_2]$, respectively. However, in general, neither the existence of isometric dilation nor the von Neumann inequality holds for $n$-tuples, $n \geq 3$, of commuting contractions. The goal of this paper is to provide a taste of the isometric dilations, the von Neumann inequality and a sharper version of von …
Let $H_m(\mathbb B)$ be the analytic functional Hilbert space on the unit ball $\mathbb B \subset \mathbb C^n$ with reproducing kernel $K_m(z,w) = (1 - \langle z,w \rangle)^{-m}$. Using algebraic operator identities we characterize those commuting row contractions $T \in L(H)^n$ on a Hilbert space $H$ that decompose into the direct sum of a spherical coisometry and copies of the multiplication tuple $M_z \in L(H_m(\mathbb B))^n$. For $m=1$, this leads to a Wold decomposition for partially isometric commuting row contractions that are regular at $z = 0$. For $m = 1 = n$, the results reduce to t…
