Let $u_1,\ldots,u_n$ be unitary operators on a Hilbert space $H$. We study the norm $$\left\|\sum^{i=n}_{i=1} u_i \otimes \bar u_i\right\|\leqno (1)$$ of the operator $\sum u_i \otimes \bar u_i$ acting on the Hilbertian tensor product $H\otimes_2 \overline H$. The main result of this note is
Theorem 1. For any $n$-tuple $u_1,\ldots, u_n$ of unitary operators in $B(H)$, we have $$2\sqrt{n-1} \le \left\|\sum^n_1 u_i \otimes \bar u_i\right\|.\leqno (6)$$ In other words, the right side of (6) is minimal exactly when $u_i = λ(g_i)$.
We establish a characterization of unitary equivalence of two bilateral operator valued weighted shifts with quasi-invertible weights by an operator of diagonal form. We also present an example of unitary equivalence between shifts defined on $\mathbb{C}^2$ which cannot be given by any unitary operator of diagonal form. The paper is concluded with investigation of unitary operators than can give unitary equivalence of bilateral operator valued weighted shifts.
We prove that there exists a rank one perturbation of a unitary operator on a complex separable infinite dimensional Hilbert space which is hypercyclic.
Let $A$ and $B$ be compact operators over a topological space $X$ and suppose that these operators are normal and have same distinct eigenvalues at each point. By obstruction theory, we establish a necessary and sufficient condition for $A$ and $B$ to be strongly unitarily equivalent. When $X=S^1$, we also give a sufficient condition for $A$ and $B$ to be approximately unitarily equivalent with some assumption on their eigenvalues.
In this paper, we investigate operator-valued frames with the structure of group-like unitary system. We show the commutant of the group-like unitary system can be characterized in terms of analysis operators associated with all the operator-valued Bessel generators and give some sufficient and necessary conditions to describe when an operator-valued frame generator admits a Parseval dual.
Recently, S. Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. Another construction was suggested later by the first and the third authors. Here, using a functional model for rank one perturbations of singular unitary operators, we give yet another construction of hypercyclic rank one perturbation of a unitary operator. In particular, we show that any Carleson set on the circle can be the spectrum of a perturbed (hypercyclic) operator.
A unitary operator $V$ and a rank $2$ operator $R$ acting on a Hilbert space $\H$ are constructed such that $V+R$ is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic.
A well-known theorem due to R. E. Curto and N. Salinas gives a necessary and sufficient condition for the unitary equivalence of commuting tuples of bounded linear operators acting on reproducing kernel Hilbert spaces. Inspired by this theorem, we obtain a different but equivalent criterion for the unitary equivalence of operators acting on reproducing kernel Hilbert spaces. As an application, we describe the structure of intertwining operator and prove that the decomposition of Cowen-Douglas operators is unique up to unitary equivalence.
This paper is concerned with a certain aspect of the spectral theory of unitary operators in a Hilbert space and its aim is to give an explicit construction of continuous functions of unitary operators. Starting from a given unitary operator we give a family of sequences of trigonometric polynomials converging weakly to the complex measures which allow us to define functions of the operator.
As applications of Kadison's Pythageorean and carpenter's theorems, the Schur-Horn theorem, and Thompson's theorem, we obtain an extension of Thompsons theorem to compact operators and use these ideas to give a characterization of diagonals of unitary operators. Thompson's mysterious inequality concerning the last terms of the diagonal and singular value sequences plays a central role.
We establish a regular sampling theory in the range of the analysis operator of a continuous frame having a unitary structure. The unitary structure is related with a unitary representation of a locally compact abelian group on a separable Hilbert space. The samples are defined by means of suitable discrete convolution systems which generalize some usual sampling settings; here regular sampling means that the samples are taken at a countable discrete subgroup.
In this paper we study truncated moment problems for $J$-self-adjoint, $J$-skew-self-adjoint and $J$-unitary operators. Conditions of the solvability are given. Some canonical solutions of the moment problems are constructed. As a by-product, some extension results for $J$-skew-symmetric and $J$-isometric operators are obtained.
We study weighted composition operators on Hilbert spaces of analytic functions on the unit ball with kernels of the form $(1-<z,w>)^{-γ}$ for $γ>0$. We find necessary and sufficient conditions for the adjoint of a weighted composition operator to be a weighted composition operator or the inverse of a weighted composition operator. We then obtain characterizations of self-adjoint and unitary weighted composition operators. Normality of these operators is also investigated.
We study pairs $(U,L_0)$, where $U$ is a unitary operator in $H$ and $L_0\subset H$ is a closed subspace, such that $$ P_{L_0}U|_{L_0}:L_0\to L_0 $$ has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.
This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen-Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.
In this paper, we shall characterize the components of the polar decomposition for an arbitrary $J$-unitary operator in a Hilbert space. This characterization has a quite different structure as that for complex symmetric and complex skew-symmetric operators. It is also shown that for a $J$-imaginary closed symmetric operator in a Hilbert space there exists a $J$-imaginary self-adjoint extension in a possibly larger Hilbert space (a linear operator $A$ in a Hilbert space $H$ is said to be $J$-imaginary if $f\in D(A)$ implies $Jf\in D(A)$ and $AJf = -JAf$, where $J$ is a conjugation on $H$).
Wiener-Hopf plus Hankel operators acting between Lebesgue spaces on the real line are studied in view of their invertibility, one sided-invertibility, Fredholm, and semi-Fredholm properties. This is done in two different cases: (i) when the Fourier symbols of the operators are unitary functions, and (ii) when the Fourier symbols are related with sectorial elements appearing in factorizations of functions originated by the Fourier symbols of the operators. The obtained result for the case (i) may be viewed as a Douglas-Sarason type theorem for Wiener-Hopf plus Hankel operators.
We systematically study various aspects of operator-valued multishifts. Beginning with basic properties, we show that the class of multishifts on the directed Cartesian product of rooted directed trees is contained in that of operator-valued multishifts. Further, we establish circularity, analyticity and wandering subspace property of these multishifts. In the rest part of the paper, we study the function theoretic behaviour of operator-valued multishifts. We determine the bounded point evaluation, reproducing kernel structure and the unitary equivalence of operator-valued multishifts with inv…
Let $U$ be a unitary operator defined on some infinite-dimensional complex Hilbert space ${\cal H}$. Under some suitable regularity assumptions, it is known that a local positive commutation relation between $U$ and an auxiliary self-adjoint operator $A$ defined on ${\cal H}$ allows to prove that the spectrum of $U$ has no singular continuous spectrum and a finite point spectrum, at least locally. We show that these conclusions still hold under weak regularity hypotheses and without any gap condition. As an application, we study the spectral properties of the Floquet operator associated to som…
The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a complex vector space, which is a generalization of the discrete-time quantum walks with constant coin matrices, are discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of the periodic unitary transition opera…
Operator model and a trace formula for pairs of unitary operators
Schmid College of Science and Technology Chapman University（查普曼大学施密德科学与技术学院）；Politecnico di Milano（米兰理工大学）；Ben-Gurion University of the Negev（内盖夫本古里安大学）
利用L. de Branges和J. Rovnyak引入的再生核希尔伯特空间理论，证明希尔伯特空间中算子对的迹公式，考虑卡拉西奥多里函数为有理函数的特殊情况，并应用于一阶离散系统理论。
An isometric operator V in a Pontryagin space H is called standard, if its domain and the range are nondegenerate subspaces in H. A description of coresolvents for standard isometric operators is known and basic underlying concepts that appear in the literature are unitary colligations and characteristic functions. In the present paper generalized coresolvents of non-standard Pontryagin space isometric operators are described. The methods used in this paper rely on a new general notion of boundary pairs introduced for isometric operators in a Pontryagin space setting. Even in the Hilbert space…
There are considered isometries on a Hilbert space. By the Wold theorem any isometry can be decomposed into a unitary operator and a unilateral shift. For a pair of isometries, even commuting, a maximal subspace reducing one isometry to a unitary operator might not reduce the other isometry. In the paper are considered pairs of commuting isometries which are completely non doubly commuting. For such pairs there are no nontrivial subspaces reducing both isometries and one of them to a unitary operator. The results describe a unitary part of an isometry in such a pair.
In this paper we introduce the concept of quadratic operator perspective for a continuous function Φ defined on the positive semi-axis of real numbers. This generalize the quadratic weighted operator geometric mean and the quadratic relative operator entropy. Some inequalities for this perspective of convex functions are established. Applications for quadratic weighted operator geometric mean and quadratic relative operator entropy are also provided.
Let $f$ be a regular non-constant symbol defined on the $d$-dimensional torus ${\mathbb T}^d$ with values on the unit circle. Denote respectively by $κ$ and $L$, its set of critical points and the associated Laurent operator on $l^2({\mathbb Z}^d)$. Let $U$ be a suitable unitary local perturbation of $L$. We show that the operator $U$ has finite point spectrum and no singular continuous component away from the set $f(κ)$. We apply these results and provide a new approach to analyze the spectral properties of GGT matrices with asymptotically constant Verblunsky coefficients. The proofs are base…
Recently, Sophie Grivaux showed that there exists a rank one perturbation of a unitary operator in a Hilbert space which is hypercyclic. We give a similar construction using a functional model for rank one perturbations of singular unitary operators.
We consider a weighted form of the Poisson summation formula. We prove that under certain decay rate conditions on the weights, there exists a unique unitary Fourier-Poisson operator which satisfies this formula. We next find the diagonal form of this operator, and prove that under weaker conditions on the weights, a unique unitary operator still exists which satisfies a Poisson summation formula in operator form. We also generalize the interplay between the Fourier transform and derivative to those Fourier-Poisson operators.
In this paper we investigate operators unitarily equivalent to truncated Toeplitz operators. We show that this class contains certain sums of tensor products of truncated Toeplitz operators. In particular, it contains arbitrary inflations of truncated Toeplitz operators; this answers a question posed by Cima, Garcia, Ross, and Wogen.
We describe a subclass of the class of normal operators on Banach spaces over non-Archimedean fields (A. N. Kochubei, J. Math. Phys. 51 (2010), article 023526) consisting of operators whose properties resemble those of unitary operators. In particular, an analog of Stone's theorem about one-parameter groups of unitary operators is proved.
In this paper, we describe families of those bounded linear operators on a separable Hilbert space that are simultaneously unitarily equivalent to integral operators on $L_2(R)$ with bounded and arbitrarily smooth Carleman kernels. The main result is a qualitative sharpening of an earlier result of [7].
In this paper, we characterize the families of those bounded linear operators on a separable Hilbert space which are simultaneously unitarily equivalent to integral bi-Carleman operators on $L_2(R)$ having arbitrarily smooth kernels of Mercer type. The main result is a qualitative sharpening of an earlier result of [7].
We study some examples of minimal length curves in homogeneous spaces of B(H) under a left action of a unitary group. Recent results relate these curves with the existence of minimal (with respect to a quotient norm) anti-Hermitian operators Z in the tangent space of the starting point. We show minimal curves that are not of this type but nevertheless can be approximated uniformly by those.
We show that (for the weak operator topology) the set of unitary operators on a separable infinite-dimensional Hilbert space is residual in the set of all contractions. The analogous result holds for isometries and the strong operator topology as well. These results are applied to the problem of embedding operators into strongly continuous semigroups.
The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of operator Lipschitz functions on the unit circle.
The main result of the paper is a description of the class of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. We prove that this class of functions coincides with the class of operator Lipschitz functions.
We present a new technique to obtain polynomial decay estimates for the matrix coefficients of unitary operators. Our approach, based on commutator methods, applies to nets of unitary operators, unitary representations of topological groups, and unitary operators given by the evolution group of a self-adjoint operator or by powers of a unitary operator. Our results are illustrated with a wide range of examples in quantum mechanics and dynamical systems, as for instance Schrödinger operators, Dirac operators, quantum waveguides, horocycle flows, adjacency matrices, Jacobi matrices, quantum walk…
In the recent preprint (OA/0006224) S. Parrott proved the equality between the Arveson`s curvature and the Fredholm index of a "pure" contraction with finite defect numbers. In the present note one derives a similar formula in the "non-pure" case.
The problems considered in this paper come as a natural continuation of our program to develop a free analogue of Sz.-Nagy-Foias theory, for row contractions. The paper is structured as follows: Introduction Part I. Unitary invariants for n-tuples of operators 1. Joint numerical radius 2. Euclidean operator radius 3. Joint numerical range and spectrum 4. $ρ$-operator radius Part II. Joint operator radii, inequalities, and applications 5. von Neumann inequalities 6. Constrained von Neumann inequalities 7. Multivariable Haagerup-de la Harpe inequalities 8. Multivariable Fejer inequalities Refere…
We use symbolic expressions for traces of positive integer powers of a Hermitian operator (or, equivalently, coefficients of corresponding characteristic polynomial) to find solutions for the problems as follows: Factorization of characteristic polynomial that collects all eigenvalues that have the same multiplicity into one polynomial factor with coefficients rationally expressed through coefficients of original characteristic polynomial. Finding with any accuracy a minimal distance between eigenvalues, as well as a minimal and a maximal eigenvalues of the Hermitian operator, applying only ra…
This paper is the second of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we give a unitary complete proof of the classification of the unitary series of the Neveu-Schwarz algebra, by the way of GKO construction, Kac determinant and FQS criterion. We then obtain the characters directly, without Feigin-Fuchs resolutions.
We will investigate the norm closure of the unitary and similarity orbits of normal operators in unital, simple, purely infinite C*-algebras. An operator theoretic proof will be given to the classification of when two normal operators are approximately unitarily equivalent in said algebras with trivial K_1-group. Some upper and lower bounds for the distance between unitary orbits will be obtained based on these methods. In addition, a complete characterization of when one normal operator is in the closed similarity orbit of another normal operator will be given for unital, simple, purely infin…
Denote by w(A) the numerical radius of a bounded linear operator A acting on Hilbert space. Suppose that A is invertible and that the numerical radius of A and of its inverse are no greater than 1+e for some non-negative e. It is shown that the distance of A from unitary operators is less or equal than a constant times $e^{1/4}$. This generalizes a result due to J.G. Stampfli, which is obtained for e = 0. An example is given showing that the exponent 1/4 is optimal. The more general case of the operator $ρ$-radius is discussed for $ρ$ between 1 and 2.
We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm - in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II_1-factors.
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local finiteness of point spectrum. Large families of locally smooth operators are also exhibited. Half of the paper is dedicated to applications, and a special emphasize is put on the study of cocycles over irrational rotations. It is apparently the first time that commutator methods are applied in the context of rotation algebras, for the study of their generator…
We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${\cal H}_0,{\cal H}$ and for an identification operator $J:{\cal H}_0\to{\cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation form…
Unlike Toeplitz operators on $H^2$, truncated Toeplitz operators do not have a natural matricial characterization. Consequently, these operators are difficult to study numerically. In this note we provide criteria for a matrix with distinct eigenvalues to be unitarily equivalent to a truncated Toeplitz operator having an analytic symbol. This test is constructive and we illustrate it with several examples. As a byproduct, we also prove that every complex symmetric operator on a Hilbert space of dimension $\leq 3$ is unitarily equivalent to a direct sum of truncated Toeplitz operators.
Let $H$ be a complex Hilbert space and let ${\mathcal C}$ be a conjugacy class of finite rank self-adjoint operators on $H$ with respect to the action of unitary operators. We suppose that ${\mathcal C}$ is formed by operators of rank $k$ and for every $A\in {\mathcal C}$ the dimensions of distinct maximal eigenspaces are distinct. Under the assumption that $\dim H\ge 4k$ we establish that every bijective transformation $f$ of ${\mathcal C}$ preserving the commutativity in both directions is induced by a unitary or anti-unitary operator, i.e. there is a unitary or anti-unitary operator $U$ suc…
