A tuple of commuting Hilbert space operators $(S_1, \dots, S_{n-1}, P)$ having the closed symmetrized polydisc \[ Γ_n = \left\{ \left(\sum_{i=1}^{n}z_i, \sum\limits_{1\leq i<j\leq n} z_iz_j, \cdots, \prod_{i=1}^{n}z_i\right) : |z_i|\leq 1\,, \; \; \; 1\leq i \leq n-1 \right\} \] as a spectral set is called a $Γ_n$-contraction. From the literature we have that a point $(s_1, \dots , s_{n-1},p)$ in $Γ_n$ can be represented as $s_i=c_i+pc_{n-i}$ for some $(c_1, \dots, c_{n-1}) \in Γ_{n-1}$. We construct a minimal $Γ_n$-isometric dilation for a particular class of c.n.u. $Γ_n$-contractions $(S_1, \cdots, S_{n-1},P)$ and obtain a functional model for them. With the help of this model we express each $S_i$ as $S_i=C_i+PC_{n-i}$, which is an operator theoretic analogue of the scalar result. We also produce an abstract model for a different class of c.n.u. $Γ_n$-contractions satisfying $S_i^*P=PS_i^*$ for each $i$. By exhibiting a counter example we show that such abstract model may not exist if we drop the hypothesis that $S_i^*P=PS_i^*$. We apply this abstract model to achieve a complete unitary invariant for such c.n.u. $Γ_n$-contractions. Additionally, we present different necessary conditions for dilation and a sufficient condition under which a commuting tuple $(S_1, \dots , S_{n-1},P)$ becomes a $Γ_n$-contraction. The entire program goes parallel to the operator theoretic program developed by Sz.-Nagy and Foias for a c.n.u. contraction.
A functional model for nondissipative unbounded perturbations of an unbounded self-adjoint operator on a Hilbert space X is constructed. This model is analogous to the Nagy--Foias model of dissipative operators, but it is linearly similar and not unitarily equivalent to the operator. It is attached to a domain of parabolic type, instead of a half-plane. The transformation map from X to the model space and the analogue of the characteristic function are given explicitly. All usual consequences of the Nagy--Foias construction (the H-infty calculus, the commutant lifting, etc.) hold true in our c…
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 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.
Using the tools of Sz.-Nagy--Foias theory of contractions, we describe in detail the invariant subspaces of the operator $ S\oplus S^* $, where $ S $ is the unilateral shift on a Hilbert space. This answers a question of Câmara and Ross.
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…
This dissertation summarizes my investigations in operator theory during my PhD studies. The first chapter is an introduction to that field of operator theory which was developed by B. Sz.-Nagy and C. Foias, the theory of power-bounded Hilbert space operators. In the second and third chapter I characterize operators which arise from power-bounded operators asymptotically. Chapter 4 is devoted to provide a possible generalization of (the necessity part of) Sz.-Nagy's famous similarity theorem. In Chapter 5 I collected my results concerning the commutant mapping of asymptotically non-vanishing c…
Using the Sz.-Nagy--Foias theory of contractions, we obtain general results about reducibility for a class of completely nonunitary contractions. These are applied to certain truncated Toeplitz operators, previously considered by Li--Yang--Lu and Gu. In particular, a negative answer is given to a conjecture stated by the latter.
We provide an explicit normal distinguished boundary dilation to a pair of commuting operators $(S,P)$ having the closed symmetrized bidisc $Γ$ as a spectral set. This is called Sz.-Nagy dilation of $(S,P)$. The operator pair that dilates $(S,P)$ is obtained by an application of Stinespring dilation of $(S,P)$ given by Agler and Young. We further prove that the dilation is minimal and the dilation space is no bigger than the dilation space of the minimal unitary dilation of the contraction $P$. We also describe model space and model operators for such a pair $(S,P)$.
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…
Sz.-Nagy's famous theorem states that a bounded operator $T$ which acts on a complex Hilbert space $\mathcal{H}$ is similar to a unitary operator if and only if $T$ is invertible and both $T$ and $T^{-1}$ are power bounded. There is an equivalent reformulation of that result which considers the self-adjoint iterates of $T$ and uses a Banach limit $L$. In this paper first we present a generalization of the necessity part in Sz.-Nagy's result concerning operators that are similar to normal operators. In the second part we provide characterization of all possible strong operator topology limits o…
Let $T$ and $V$ be two Hilbert space contractions and let $X$ be a linear bounded operator. It was proved by C. Foias and J.P. Williams that in certain cases the operator block matrix $R(X;T,V)$ (defined in the text) is similar to a contraction if and only if the commutator equation $X = TZ-ZV$ has a bounded solution $Z$. We characterize here the similarity to contractions of some operator matrices $R(X;T,V)$ in terms of growth conditions or of perturbations of $R(0;T,V)=T\oplus V$.
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…
The purpose of this paper is to describe the Harnack parts for the operators of class C $ρ$ ($ρ$ \textgreater{} 0) on Hilbert spaces which were introduced by B. Sz. Nagy and C. Foias in [25]. More precisely, we study Harnack parts of operators with $ρ$-numerical radius one. The case of operators with $ρ$-numerical radius strictly less than 1 was described in [10]. We obtain a general criterion for compact $ρ$-contractions to be in the same Harnack part. We give a useful equivalent form of this criterion for usual contractions. Operators with numerical radius one received also a particular atte…
The principal theorem of Sz.-Nagy on dilation of a positive definite Hilbert space operator valued function has played a central role in the development of the non-self-adjoint operator theory. In this paper we introduce the positive definiteness for locally Hilbert space operator valued kernels, we prove an analogue of the Sz.-Nagy dilation theorem and, as application, we obtain dilation results for locally contractions and locally $ρ$ - contractions as well as for locally semi-spectral measures.
The explicit constructions of minimal isometric, and minimal unitary dilations of an arbitrary linear pencil of operators $T(λ)=T_0+λT_1$ consisting of contractions on a separable Hilbert space for $|λ|=1$, which generalize the classical constructions (the case $T_1=0$), are presented. In contrast to the classical case these dilations are essentially non-unique.
A particular case of results from [K2] is as follows. Let the unitary asymptote of a contraction $T$ contain the bilateral shift (of finite or infinite multiplicity). Then there exists an invariant subspace $\mathcal M$ of $T$ such that $T|_{\mathcal M}$ is similar to the unilateral shift of the same multiplicity. The proof is based on the Sz.-Nagy--Foias functional model for contractions. In the present paper this result is generalized to polynomially bounded operators, but in the simplest case. Namely, it is proved that if the unitary asymptote of a polynomially bounded operator $T$ contains…
Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ α[T^*,T]:=\sum_{n=0}^\infty α_n T^{*n}T^n\ge 0. \] where $α(t)=\sum_{n=0}^\infty α_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $α(t) = (1-t) \tildeα (t)$, where $\tildeα$ has no roots in $[0,1]$, then $T$ is similar to a contraction. Operators of this type have been investigated by Agler, Müller, Olofsson, Pott and others, however, we treat cases where their techniques do not apply. We write down an explicit Nagy-Foias type model of an operator in this clas…
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…
A classical result of Sz.-Nagy asserts that a Hilbert-space contraction operator $T$ can be lifted to an isometry $V$. A more general multivariable setting of recent interest for these ideas is the case where (i) the unit disk is replaced by a certain domain contained in ${\mathbb C}^3$ (called the {\em tetrablock}), (ii) the contraction operator $T$ is replaced by a commutative triple $(T_1, T_2, T)$ of Hilbert-space operators having ${\mathbb E}$ as a spectral set (a tetrablock contraction) . The rational dilation question for this setting is whether a tetrablock contraction $(T_1, T_2, T)$ …
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…
One can view contraction operators given by a canonical model of Sz.-Nagy and Foias as being defined by a quotient module where the basic building blocks are Hardy spaces. In this note we generalize this framework to allow the Bergman and weighted Bergman spaces as building blocks, but restricting attention to the case in which the operator obtained is in the Cowen-Douglas class and requiring the multiplicity to be one. We view the classification of such operators in the context of complex geometry and obtain a complete classification up to unitary equivalence of them in terms of their associa…
In this paper we develop the method of double operator integrals to prove trace formulae for functions of contractions, dissipative operators, unitary operators and self-adjoint operators. To establish the absolute continuity of spectral shift, we use the Sz.-Nagy theorem on the absolute continuity of the spectrum of the minimal unitary dilation of a completely nonunitary contraction. We also give a construction of an intermediate contraction for a pair of contractions with trace class difference.
We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V…
This survey aims to give a brief introduction to operator theory in the Hardy space over the bidisc $H^2(\mathbb D^2)$. As an important component of multivariable operator theory, the theory in $H^2(\mathbb D^2)$ focuses primarily on two pairs of commuting operators that are naturally associated with invariant subspaces (or submodules) in $H^2(\mathbb D^2)$. Connection between operator-theoretic properties of the pairs and the structure of the invariant subspaces is the main subject. The theory in $H^2(\mathbb D^2)$ is motivated by and still tightly related to several other influential theorie…
The goal of the present paper is to introduce and study noncommutative Hardy spaces associated with the regular $Λ$-polyball, to develop a functional calculus on noncommutative Hardy spaces for the completely non-coisometric (c.n.c.) $k$-tuples in ${\bf B}_Λ(H)$, and to study the characteristic functions and the associated multi-analytic models for the c.n.c. elements in the regular $Λ$-polyball. In addition, we show that the characteristic function is a complete unitary invariant for the class of c.n.c. $k$-tuples in ${\bf B}_Λ(H)$. These results extend the corresponding classical results of …
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 …
We suggest a new version of the notion of $ρ$-dilation ($ρ>0$) of an $N$-tuple $\mathbf{A}=(A_1,...,A_N)$ of bounded linear operators on a common Hilbert space. We say that $\mathbf{A}$ belongs to the class $C_{ρ,N}$ if $\mathbf{A}$ admits a $ρ$-dilation $\widetilde{\mathbf{A}}=(\widetilde{A}_1,...,\widetilde{A}_N)$ for which $ζ\widetilde{\mathbf{A}}:=ζ_1\widetilde{A}_1+... +ζ_N\widetilde{A}_N$ is a unitary operator for each $ζ:=(ζ_1,...,ζ_N)$ in the unit torus $\mathbb{T}^N$. For N=1 this class coincides with the class $C_ρ$ of B. Sz.-Nagy and C. Foiaş. We generalize the known descriptions of…
A classical result of Sz.-Nagy asserts that a Hilbert space contraction operator $T$ can be dilated to a unitary $\cU$. A more general multivariable setting for these ideas is the setup where (i) the unit disk is replaced by a domain $Ω$ contained in ${\mathbb C}^d$, (ii) the contraction operator $T$ is replaced by a commuting tuple $\bfT = (T_1, \dots, T_d)$ such that $\| r(T_1, \dots, T_d) \|_{\cL(\cH)} \le \sup_{\lam \in Ω} | r(\lam) |$ for all rational functions with no singularities in $\overlineΩ$ and the unitary operator $\cU$ is replaced by an $Ω$-unitary operator tuple, i.e., a commut…
The main issue we address in the present paper are the new models for completely non-unitary contractions with rank one defect operators acting on some Hilbert space of dimension $N\leq\infty$. This model complements nicely the well-known models of Liv$\rm{\check{s}}$ic and Sz.-Nagy--Foias. We show that each such an operator is unitarily equivalent to some truncated CMV matrix obtained from the ``full'' CMV matrix by deleting the first row and the first column, and acting in $\ell^2$ ($\dC^N$). This result can be viewed as a nonunitary version of the famous characterization of unitary operator…
A commuting pair of operators (S, P) on a Hilbert space H is said to be a Gamma-contraction if the symmetrized bidisc is a spectral set of the tuple (S, P). In this paper we develop some operator theory inspired by Agler and Young's results on a model theory for Gamma-contractions. We prove a Beurling-Lax-Halmos type theorem for Gamma-isometries. Along the way we solve a problem in the classical one-variable operator theory. We use a "pull back" technique to prove that a completely non-unitary Gamma-contraction (S, P) can be dilated to a direct sum of a Gamma-isometry and a Gamma-unitary on th…
The celebrated theorem of Berger, Coburn and Lebow on pairs of commuting isometries can be formulated as follows: a pure isometry $V$ on a Hilbert space $\mathcal{H}$ is a product of two commuting isometries $V_1$ and $V_2$ in $\mathcal{B}(\mathcal{H})$ if and only if there exists a Hilbert space $\mathcal{E}$, a unitary $U$ in $\mathcal{B}(\mathcal{E})$ and an orthogonal projection $P$ in $\mathcal{B}(\mathcal{E})$ such that $(V, V_1, V_2)$ and $(M_z, M_Φ, M_Ψ)$ on $H^2_{\mathcal{E}}(\mathbb{D})$ are unitarily equivalent, where \[ Φ(z)=(P+zP^{\perp})U^*\;\text{and}\; Ψ(z)=U(P^{\perp}+zP) \;;(…
A commuting tuple of operators $(S_1,\dots, S_{n-1},P)$, defined on a Hilbert space $\mathcal H$, for which the closed symmetrized polydisc \[ Γ_n =\left\{ \left(\sum_{1\leq i\leq n} z_i,\sum_{1\leq i<j\leq n}z_iz_j,\dots, \prod_{i=1}^n z_i \right): \,|z_i|\leq 1, i=1,\dots,n \right \} \] is a spectral set, is called a $Γ_n$-\textit{contraction}. A $Γ_n$-contraction is said to be \textit{pure} or $C._0$ if $P$ is $C._0$, that is, if ${P^*}^n \rightarrow 0$ strongly as $n \rightarrow \infty$. We show that for any $Γ_n$-contraction $(S_1,\dots, S_{n-1},P)$, there is a unique operator tuple $(A_1…
For a unitary operator the family of its unitary perturbations by rank one operators with fixed range is parametrized by a complex parameter $γ, |γ|=1$. Namely all such unitary perturbations are $U_γ:=U+(γ-1) (., b_1)_{\mathcal H} b$, where $b\in\mathcal H, \|b\|=1, b_1=U^{-1} b, |γ|=1$. For $|γ|<1$ operators $U_γ$ are contractions with one-dimensional defects. Restricting our attention on the non-trivial part of perturbation we assume that $b$ is cyclic for $U$. Then the operator $U_γ$, $|γ|<1$ is a completely non-unitary contraction, and thus unitarily equivalent to its functional model $\ma…
