We consider a family of norms (called operator E-norms) on the algebra $B(H)$ of all bounded operators on a separable Hilbert space $H$ induced by a positive densely defined operator $G$ on $H$. Each norm of this family produces the same topology on $B(H)$ depending on $G$. By choosing different generating operator $G$ one can obtain operator E-norms producing different topologies, in particular, the strong operator topology on bounded subsets of $B(H)$. We obtain a generalised version of the Kretschmann-Schlingemann-Werner theorem, which shows continuity of the Stinespring representation of CP linear maps w.r.t. the energy-constrained $cb$-norm (diamond norm) on the set of CP linear maps and the operator E-norm on the set of Stinespring operators. The operator E-norms induced by a positive operator $G$ are well defined for linear operators relatively bounded w.r.t. the operator $\sqrt{G}$ and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between the operator E-norms and the standard characteristics of $\sqrt{G}$-bounded operators. The operator E-norms allow to obtain simple upper bounds and continuity bounds for
We establish operator-valued versions of the earlier foundational factorization results for noncommutative polynomials due to Helton (Ann.~Math., 2002) and one of the authors (Linear Alg.~Appl., 2001). Specifically, we show that every positive operator-valued noncommutative polynomial $p$ admits a single-square factorization $p=r^{*}r$. An analogous statement holds for operator-valued noncommutative trigonometric polynomials. Our approach follows the now standard sum-of-squares (sos) paradigm but requires new results and constructions tailored to operator coefficients. Assuming a positive $p$ is not sos, Hahn--Banach separation yields a linear functional that is positive on the sos cone and negative on $p$; a Gelfand--Naimark--Segal (GNS) construction then produces a representing tuple $Y$ leading to contradiction since $p$ was assumed positive on $Y$. The main technical input is a canonical tuple $A$ of self-adjoint operators and, in the unitary case, a canonical tuple $U$ of unitaries, both constructed from the left-regular representation on Fock space. We prove that, up to a universal constant, the norms $\|p(A)\|$ and $\|p(U)\|$ bound the operator norm of any positive semidefin
A regular operator T on a Hilbert C^*-module is defined just like a closed operator on a Hilbert space, with the extra condition that the range of (I+T^*T) is dense. Semiregular operators are a slightly larger class of operators that may not have this property. It is shown that, like in the case of regular operators, one can, without any loss in generality, restrict oneself to semiregular operators on C^*-algebras. We then prove that for abelian C^*-algebras as well as for subalgebras of the algebra of compact operators, any closed semiregular operator is automatically regular. We also determine how a regular operator and its extensions (and restrictions) are related. Finally, using these results, we give a criterion for a semiregular operator on a liminal C^*-algebra to have a regular extension.
We exhibit a general class of unbounded operators in Banach spaces which can be shown to have the single-valued extension property, and for which the local spectrum at suitable points can be determined. We show that a local spectral radius formula holds, analogous to that for a globally defined bounded operator on a Banach space with the single-valued extension property. An operator of the class under consideration can occur in practice as (an extension of) a differential operator which, roughly speaking, can be diagonalised on its domain of smooth test functions via a discrete transform, such that the diagonalising transform establishes an isomorphism of topological vector spaces between the domain of the differential operator, in its own topology, and a sequence space. We give concrete examples of (extensions of) such operators (constant coefficient differential operators on the d-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.
In this notes unbounded regular operators on Hilbert $C^*$-modules over arbitrary $C^*$-algebras are discussed. A densely defined operator $t$ possesses an adjoint operator if the graph of $t$ is an orthogonal summand. Moreover, for a densely defined operator $t$ the graph of $t$ is orthogonally complemented and the range of $P_FP_{G(t)^\bot}$ is dense in its biorthogonal complement if and only if $t$ is regular. For a given $C^*$-algebra $\mathcal A$ any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules is regular, if and only if any densely defined $\mathcal A$-linear closed operator $t$ between Hilbert $C^*$-modules admits a densely defined adjoint operator, if and only if $\mathcal A$ is a $C^*$-algebra of compact operators. Some further characterizations of closed and regular modular operators are obtained. Changes 1: Improved results, corrected misprints, added references. Accepted by J. Operator Theory, August 2007 / Changes 2: Filled gap in the proof of Thm. 3.1, changes in the formulations of Cor. 3.2 and Thm. 3.4, updated references and address of the second author.
The purpose of this paper is to introduce a consistent notion of universal and reduced crossed products by actions and coactions of groups on operator systems and operator spaces. In particular we shall put emphasis to reveal the full power of the universal properties of the the universal crossed products. It turns out that to make things consistent, it seems useful to perform our constructions on some bigger categories which allow the right framework for studying the universal properties and which are stable under the construction of crossed products even for non-discrete groups. In the case of operator systems, this larger category is what we call a $C^*$-operator system, i.e., a selfadjoint subspace $X$ of some $\mathcal B(H)$ which contains a $C^*$-algebra $A$ such that $AX=X=XA$. In the case of operator spaces, the larger category is given by what we call $C^*$-operator bimodules. After we introduced the respective crossed products we show that the classical Imai-Takai and Katayama duality theorems for crossed products by group (co-)actions on $C^*$-algebras extend one-to-one to our notion of crossed products by $C^*$-operator systems and $C^*$-operator bimodules.
The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the $P$ representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the $P$ representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the $P$ representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter $s$ assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for $\mathrm{Res}<0$, has bounded eigenvalues for $\mathrm{Res}=0$, and has infinite eigenvalues for $s=1$. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter $s$ is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to $s=+1$, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the $P$ representation is appropriate.
An operator *-algebra is a non-selfadjoint operator algebra with completely isometric involution. We show that any operator *-algebra admits a faithful representation on a Hilbert space in such a way that the involution coincides with the operator adjoint up to conjugation by a symmetry. We introduce operator *-correspondences as a general class of inner product modules over operator *-algebras and prove a similar representation theorem for them. From this we derive the existence of linking operator *-algebras for operator *-correspondences. We illustrate the relevance of this class of inner product modules by providing numerous examples arising from noncommutative geometry.
For a locally compact group $G$ we consider the algebra $CD(G)$ of convolution dominated operators on $L^{2}(G)$: An operator $A:L^2(G)\to L^2(G)$ is called convolution dominated if there exists $a\in L^1(G)$ such that for all $f \in L^2(G)$ $ |Af(x)| \leq a * |f| (x)$, for almost all $x \in G$. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operators $CD_{reg}(G)$. For amenable $G$ which is rigidly symmetric as a discrete group, we show that any element of $CD_{reg}(G)$ is invertible in $CD_{reg}(G)$ if it is invertible as a bounded operator on $L^2(G)$. We give an example of a symmetric group $E$ for which the convolution dominated operators are not inverse-closed in the bounded operators on $L^2(E)$.
Let $\varphi$ be a holomorphic self map of the bidisc that is Lipschitz on the closure. We show that the composition operator $C_{\varphi}$ is compact on the Bergman space if and only if $\varphi(\overline{\mathbb{D}^2})\cap \mathbb{T}^2=\emptyset$ and $\varphi(\overline{\mathbb{D}^2}\setminus \mathbb{T}^2)\cap b\mathbb{D}^2=\emptyset$.
Let B be a unital C*-subalgebra of a unital C*-algebra A, so that A/B is an abstract operator space. We show how to realize A/B as a concrete operator space by means of a completely contractive map from A into the algebra of operators on a Hilbert space, of the form a maps to [z, a] where z is a Hermitian unitary operator. We do not use Ruan's theorem concerning concrete realization of abstract operator spaces. Along the way we obtain corresponding results for abstract operator spaces of the form A/V where V is a closed subspace of A, and then for the more special cases in which V is a *-subspace or an operator system.
We develop important properties of the KK-functor on the basis of split exactness. In particular we discuss two slightly different short proofs for the existence of the Kasparov product and its associativity. We use the approach with quasihomomorphisms to obtain a short proof of the fact that operator homotopy implies homotopy . Using an idea of Gabe-Szabo we also deduce from this the 'stable uniqueness theorem' of Dadarlat-Eilers by a very short argument.
We introduce an analogue for Lip-normed operator systems of the second author's order-unit quantum Gromov-Hausdorff distance and prove that it is equal to the first author's complete distance. This enables us to consolidate the basic theory of what might be called operator Gromov-Hausdorff convergence. In particular we establish a completeness theorem and deduce continuity in quantum tori, Berezin-Toeplitz quantizations, and theta-deformations from work of the second author. We show that approximability by Lip-normed matrix algebras is equivalent to 1-exactness of the underlying operator space and, by applying a result of Junge and Pisier, that for n greater than or equal to 7 the set of isometry classes of n-dimensional Lip-normed operator systems is nonseparable. We also treat the question of generic complete order structure.
Let $Ω_+$ be either the open unit disc or the open upper half plane or the open right half plane. In this paper, we compute the norm of the basic operator $A_α=Π_ΘT_{b_α}|_{\mathcal{H}(Θ)}$ in the vector valued model space $\mathcal{H}(Θ)=H^m_2 \ominus ΘH^m_2$ associated with an $m\times m$ matrix valued inner function $Θ$ in $Ω_+$ and show that the norm is attained. Here $Π_Θ$ denotes the orthogonal projection from the Lebesgue space $L^m_2$ onto $\mathcal{H}(Θ)$ and $T_{b_α}$ is the operator of multiplication by the elementary Blaschke factor $b_α$ of degree one with a zero at a point $α\in Ω_+$. We show that if $A_α$ is strictly contractive, then its norm may be expressed in terms of the singular values of $Θ(α)$. We then extend this evaluation to the more general setting of vector valued de Branges spaces.
1 Homogenization of Second Order Elliptic Operators with Periodic Coefficients.- 1.1 Preliminaries.- 1.2 Setting of the Homogenization Problem.- 1.3 Problems of Justification Further Examples.- 1.4 The Method of Asymptotic Expansions.- 1.5 Explicit Formulas for the Homogenized Matrix in the Two-Dimensional Case.- 1.6 Estimates and Approximations for the Homogenized Matrix.- 1.7 The Rayleigh-Maxwell Formulas.- Comments.- 2 An Introduction to the Problems of Diffusion.- 2.1 Homogenization of Parabolic Operators.- 2.2 Homogenization and the Central Limit Theorem.- 2.3 Stabilization of Solutions of Parabolic Equations.- 2.4 Diffusion in a Solenoidal Flow.- 2.5 Diffusion in an Arbitrary Periodic Flow.- 2.6 Spectral Approach to the Asymptotic Problems of Diffusion.- 2.7 Diffusion with Absorption.- Comments.- 3 Elementary Soft and Stiff Problems.- 3.1 Homogenization of Soft Inclusions.- 3.2 Homogenization of Stiff Inclusions.- 3.3 Virtual Mass.- 3.4 The Method of Asymptotic Expansions.- 3.5 On a Dense Cubic Packing of Balls.- 3.6 The Dirichlet Problem in a Perforated Domain.- Comments.- 4 Homogenization of Maxwell Equations.- 4.1 Preliminary Results.- 4.2 A Lemma on Compensated Compactness.- 4.3 Homogenization.- 4.4 The Problem of an Artificial Dielectric.- Comments.- 5 G-Convergence of Differential Operators.- 5.1 Basic Properties of G-Convergence.- 5.2 A Sufficient Condition of G-Convergence.- 5.3 G-Convergence of Abstract Operators.- 5.4 Compactness Theorem and Its Implications.- 5.5 G-Convergence and Duality.- 5.6 Stratified Media.- 5.7 G-Convergence of Divergent Elliptic Operators of Higher Order.- Comments.- 6 Estimates for the Homogenized Matrix.- 6.1 The Hashin-Shtrikman Bounds.- 6.2 Attainability of Bounds. The Hashin Structure.- 6.3 Extremum Principles.- 6.4 The Variational Method.- 6.5 G-Limit Media Attainment of the Bounds on Stratified Composites.- 6.6 The Method of Quasi-Convexity.- 6.7 The Method of Null Lagrangians.- 6.8 The Method of Integral Representation.- Comments.- 7 Homogenization of Elliptic Operators with Random Coefficients.- 7.1 Probabilistic Description of Non-Homogeneous Media.- 7.2 Homogenization.- 7.3 Explicit Formulas in Two-Dimensional Problems.- 7.4 Homogenization of Almost-Periodic Operators.- 7.5 The General Theorem of Individual Homogenization.- Comments.- 8 Homogenization in Perforated Random Domains.- 8.1 Homogenization.- 8.2 Remarks on Positive Definiteness of the Homogenized Matrix.- 8.3 Central Limit Theorem.- 8.4 Disperse Media.- 8.5 Criterion of Pointwise Stabilization A Refinement of the Central Limit Theorem.- 8.6 Stiff Problem for a Random Spherical Structure.- 8.7 Random Spherical Structure with Small Concentration.- Comments.- 9 Homogenization and Percolation.- 9.1 Existence of the Effective Conductivity.- 9.2 Random Structure of Chess-Board Type.- 9.3 The Method of Percolation Channels.- 9.4 Conductivity Threshold for a Random Cubic Structure in ?3.- 9.5 Resistance Threshold for a Random Cubic Structure in ?3.- 9.6 Central Limit Theorem for Random Motion in an Infinite Two-Dimensional Cluster.- Comments.- 10 Some Asymptotic Problems for a Non-Divergent Parabolic Equation with Random Stationary Coefficients.- 10.1 Preliminary Remarks.- 10.2 Auxiliary Equation A*p = 0 on a Probability Space.- 10.3 Homogenization and the Central Limit Theorem.- 10.4 Criterion of Pointwise Stabilization.- Comments.- 11 Spectral Problems in Homogenization Theory.- 11.1 Spectral Properties of Abstract Operators Forming a Sequence.- 11.2 On the Spectrum of G-Convergent Operators.- 11.3 The Sturm-Liouville Problem.- 11.4 Spectral Properties of Stratified Media.- 11.5 Density of States for Random Elliptic Operators.- 11.6 Asymptotics of the Density of States.- Comments.- 12 Homogenization in Linear Elasticity.- 12.1 Some General Facts from the Theory of Elasticity.- 12.2 G-Convergence of Elasticity Tensors.- 12.3 Homogenization of Periodic and Random Tensors.- 12.4 Fourth Order Operators.- 12.5 Linear Problems of Incompressible Elasticity.- 12.6 Explicit Formulas for Two-Dimensional Incompressible Composites.- 12.7 Some Questions of Analysis on a Probability Space.- 13 Estimates for the Homogenized Elasticity Tensor.- 13.1 Basic Estimates.- 13.2 The Variational Method.- 13.3 Two-Phase Media Attainability of Bounds on Stratified Composites.- 13.4 On the Hashin Structure.- 13.5 Disperse Media with Inclusions of Small Concentration.- 13.6 Fourth Order Operators Systems of Stokes Type.- Comments.- 14 Elements of the Duality Theory.- 14.1 Convex Functions.- 14.2 Integral Functionals.- 14.3 On Two Types of Boundary Value Problems.- 14.4 Dual Boundary Value Problems.- 14.5 Extremal Relations.- 14.6 Examples of Regular Lagrangians.- Comments.- 15 Homogenization of Nonlinear Variational Problems.- 15.1 Random Lagrangians.- 15.2 Two Principal Lemmas.- 15.3 Homogenization Theorems.- 15.4 Applications to Boundary Value Problems in Perforated Domains.- 15.5 Chess Lagrangians Dychne's Formula.- Comments.- 16 Passing to the Limit in Nonlinear Variational Problems.- 16.1 Definition of ?-Convergence of Lagrangians Formulation of the Compactness Theorems.- 16.2 Convergence of Energies and Minimizers.- 16.3 Proof of the Compactness Theorems.- 16.4 Two Examples: Ulam's Problem Homogenization Problem.- 16.5 Compactness of Lagrangians in Plasticity Problems Application to Ll-Closedness.- 16.6 Remarks on Non-Convex Functionals.- Comments.- 17 Basic Properties of Abstract ?-Convergence.- 17.1 ?-Convergence of Functions on a Metric Space.- 17.2 ?-Convergence of Functions Defined in a Banach Space.- 17.3 ?-Convergence of Integral Functionals.- Comments.- 18 Limit Load.- 18.1 The Notion of Limit Load.- 18.2 Dual Definition of Limit Load.- 18.3 Equivalence Principle.- 18.4 Convergence of Limit Loads in Homogenization Problems.- 18.5 Surface Loads.- 18.6 Representation of the Functional $$\bar F$$ on BV0.- 18.7 ?-Convergence in BV0.- Comments.- Appendix A. Proof of the Nash-Aronson Estimate.- Appendix C. A Property of Bounded Lipschitz Domains.- References.
We show that the symmetrized product $AB+BA$ of two positive operators $A$ and $B$ is positive if and only if $f(A+B)\leq f(A)+f(B)$ for all non-negative operator monotone functions $f$ on $[0,\infty)$ and deduce an operator inequality. We also give a necessary and sufficient condition for that the composition $f\circ g$ of an operator convex function $f$ on $[0,\infty)$ and a non-negative operator monotone function $g$ on an interval $(a,b)$ is operator monotone and give some applications.
We show that the set of projections in an operator system can be detected using only the abstract data of the operator system. Specifically, we show that if $p$ is a positive contraction in an operator system $V$ which satisfies certain order-theoretic conditions, then there exists a complete order embedding of $V$ into $B(H)$ mapping $p$ to a projection operator. Moreover, every abstract projection in an operator system $V$ is an honest projection in the C*-envelope of $V$. Using this characterization, we provide an abstract characterization for operator systems spanned by two commuting families of projection-valued measures and discuss applications in quantum information theory.
A commuting $n$-tuple $(T_1, \ldots, T_n)$ of bounded linear operators on a Hilbert space $\clh$ associate a Hilbert module $\mathcal{H}$ over $\mathbb{C}[z_1, \ldots, z_n]$ in the following sense: \[\mathbb{C}[z_1, \ldots, z_n] \times \mathcal{H} \rightarrow \mathcal{H}, \quad \quad (p, h) \mapsto p(T_1, \ldots, T_n)h,\]where $p \in \mathbb{C}[z_1, \ldots, z_n]$ and $h \in \mathcal{H}$. A companion survey provides an introduction to the theory of Hilbert modules and some (Hilbert) module point of view to multivariable operator theory. The purpose of this survey is to emphasize algebraic and geometric aspects of Hilbert module approach to operator theory and to survey several applications of the theory of Hilbert modules in multivariable operator theory. The topics which are studied include: generalized canonical models and Cowen-Douglas class, dilations and factorization of reproducing kernel Hilbert spaces, a class of simple submodules and quotient modules of the Hardy modules over polydisc, commutant lifting theorem, similarity and free Hilbert modules, left invertible multipliers, inner resolutions, essentially normal Hilbert modules, localizations of free resolutions and rigid
The present article is a review of recent developments concerning the notion of Følner sequences both in operator theory and operator algebras. We also give a new direct proof that any essentially normal operator has an increasing Følner sequence $\{P_n\}$ of non-zero finite rank projections that strongly converges to 1. The proof is based on Brown-Douglas-Fillmore theory. We use Følner sequences to analyze the class of finite operators introduced by Williams in 1970. In the second part of this article we examine a procedure of approximating any amenable trace on a unital and separable C*-algebra by tracial states $\mathrm{Tr}(\cdot P_n)/\mathrm{Tr}(P_n)$ corresponding to a Følner sequence and apply this method to improve spectral approximation results due to Arveson and Bédos. The article concludes with the analysis of C*-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. We give an abstract characterization of these algebras in terms of unital completely positive maps and define Følner C*-algebras as those unital separable C*-algebras that satisfy these equivalent conditions. This is analogous to Voiculescu's abstra
The weighted geometric (WG) operator and the ordered weighted geometric (OWG) operator are two common aggregation operators in the field of information fusion. But these two aggregation operators are usually used in situations where the given arguments are expressed as crisp numbers or linguistic values. In this paper, we develop some new geometric aggregation operators, such as the intuitionistic fuzzy weighted geometric (IFWG) operator, the intuitionistic fuzzy ordered weighted geometric (IFOWG) operator, and the intuitionistic fuzzy hybrid geometric (IFHG) operator, which extend the WG and OWG operators to accommodate the environment in which the given arguments are intuitionistic fuzzy sets which are characterized by a membership function and a non-membership function. Some numerical examples are given to illustrate the developed operators. Finally, we give an application of the IFHG operator to multiple attribute decision making based on intuitionistic fuzzy sets.