Positive Self-adjoint Research Articles

Inequalities for the extended positive part of a von Neumann algebra related to operator-monotone and operator-convex functions

We extend inequalities for operator monotone and operator convex functions onto elements of the extended positive part of a von Neumann algebra. In particular, this provides an opportunity to extend the inequalities onto unbounded positive self-adjoint operators.

On the Energy-Momentum Spectrum and One-Meson Dispersion Curves in (3 + 1)-Dimensional Strongly Coupled Lattice QCD with three Flavors

We use an imaginary-time formulation of quantum field theory to analyze a lattice QCD model with the improved Wilson action on a Euclidean (3 + 1)-dimensional unit hypercubic lattice, with three quark flavors, local SU(3)c gauge symmetry and global SU(3)f flavor symmetry. We work in the strong coupling regime, with a small hopping parameter κ > 0 and a much smaller plaquette coupling β > 0. This work extends our previous results on the pointwisely determined one-meson dispersion curves in the low-lying one-particle energy-momentum spectrum of strongly coupled lattice QCD models. Here, we give a smoothness characterization of the dispersion curves and provide bounds on the more than one-particle contributions to the spectral measure for the two-meson correlations. Denoting by H the underlying quantum mechanical physical Hilbert space, constructed according to the Osterwalder-Seiler prescription, our spectral results are derived from spectral representations for the positive self-adjoint energy and self-adjoint momentum operators in H. After establishing a lattice Feynman-Kac formula, these representations are used to rigorously identify complex momentum singularities of the lattice Fourier transform of special two-particle correlations with energy-momentum spectral points and identify the meson particles with isolated dispersion curves. To show isolated dispersion curves we must show both lower and upper spectral gaps. To determine the eightfold-way mesons, the usual spacetime and internal symmetries are heavily used to block decompose the lattice Fourier transform of the two-meson correlation matrix. Then, to solve the implicit equations determining the dispersion curves, we apply the classical analytic implicit function theorem to the blocks associated with total spin zero mesons and the Weierstrass preparation theorem to the blocks corresponding to total spin one. The Weierstrass preparation theorem is the natural tool to deal with the determination of the spectrum when degeneracies are present and the hypotheses for the analytic implicit function theorem are not fulfilled. All the 36 meson dispersion curves are of the form, for β = 0, w(p→)=−2lnκ+r(κ,p→), with r(κ,p→) of order κ2, giving an asymptotic mass -2 ln κ. A mass splitting of 2κ4 between the pseudoscalar and vector mesons occurs at order κ4. We determine smoothness properties of the functions r(κ,p→). The results for β = 0 are extended to small β > 0 using β-analyticity properties of correlations in the infinite-volume limit.

Fractal dimensions of spectral measures of rank one perturbations of a positive self-adjoint operator

Let H be a Hilbert space. Suppose A is a positive self-adjoint operator on H and φ∈H is a cyclic unit vector. For each λ∈R, we can define the rank one perturbation of A by Aλ=A+λ〈φ,⋅〉φ. To each Aλ we can consider the spectral measure of φ, which we denote by μλ. This generates a family of measures, {μλ}, and we analyze the packing dimension of this family. Past results have determined that the Hausdorff dimension of this family can be determined if the limit inferior of a ratio involving μ is constant on a Lebesgue typical set. This ratio is sometimes called the pointwise dimension of μ and is related to the upper derivative of μ. Work has been done to make a similar argument for the packing dimension, but with little success. Using the theory of rank one perturbations and Borel transforms, we introduce the concept of Lebesgue exact dimension for μ, which allows us to determine the packing dimension of spectral measures of almost every rank one perturbation μλ. If the Lebesgue exact dimension for μ is 1<α<2 then the packing dimension of Lebesgue almost every μλ is 2−α. As a corollary, we find that this limit condition implies a stronger result: the Hausdorff and packing dimensions are equal for almost every μλ.

Entropy Distribution of Localised States

We study the geometric distribution of the relative entropy of a charged localised state in Quantum Field Theory. With respect to translations, the second derivative of the vacuum relative entropy is zero out of the charge localisation support and positive in mean over the support of any single charge. For a spatial strip, the asymptotic mean entropy density is $\pi E$, with $E$ the corresponding vacuum charge energy. In a conformal QFT, for a charge in a ball of radius $r$, the relative entropy is non linear, the asymptotic mean radial entropy density is $\pi E$ and Bekenstein's bound is satisfied. We also study the null deformation case. We construct, operator algebraically, a positive selfadjoint operator that may be interpreted as the deformation generator, we thus get a rigorous form of the Averaged Null Energy Condition that holds in full generality. In the one dimensional conformal $U(1)$-current model, we give a complete and explicit description of the entropy distribution of a localised charged state in all points of the real line; in particular, the second derivative of the relative entropy is strictly positive in all points where the charge density is non zero, thus the Quantum Null Energy Condition holds here for these states and is not saturated in these points.

On the ultimate energy bound of solutions to some forced second-order evolution equations with a general nonlinear damping operator

Under suitable growth and coercivity conditions on the nonlinear damping operator $g$ which ensure non-resonance, we estimate the ultimate bound of the energy of the general solution to the equation $\ddot{u}(t) + Au(t) + g(\dot{u}(t))=h(t),\quad t\in\mathbb{R}^+ ,$ where $A$ is a positive selfadjoint operator on a Hilbert space $H$ and $h$ is a bounded forcing term with values in $H$. In general the bound is of the form $ C(1+ ||h||^4)$ where $||h||$ stands for the $L^\infty$ norm of $h$ with values in $H$ and the growth of $g$ does not seem to play any role. If $g$ behaves lie a power for large values of the velocity, the ultimate bound has a quadratic growth with respect to $||h||$ and this result is optimal. If $h$ is anti periodic, we obtain a much lower growth bound and again the result is shown to be optimal even for scalar ODEs.

Second-order difference approximations for Volterra equations with the completely monotonic kernels

The second-order difference type methods are studied for the solution of the problem $$u^{\prime}(t)+{{\int}_{0}^{t}} (t-\tau)u (\tau)d\tau = 0 , t>0, u(0) = u_{0}, $$ with $ L(t)=\sum \limits _{j = 1}^{n} a_{j}(t) L_{j} $ . The operators Lj are densely defined positive self-adjoint linear operator on a Hilbert space H and have spectral decompositions with respect to a common resolution of the identity {Eλ} in H. Here, the kernel functions aj(t), 1 ≤ j ≤ n, are completely monotonic on (0, ∞) and locally integrable, but not constant. The convergence properties of the time discretization are proven in the weighted $ l^{1}(\rho ;0,\infty ; \mathbf {H} ) $ and $ l^{\infty }(\rho ; 0, \infty ; \mathbf {H} ) $ norm, where ρ is a given weighted function.

Backward semi-linear parabolic equations with time-dependent coefficients and local Lipschitz source

Let H be a Hilbert space with the inner product and the norm , a positive self-adjoint unbounded time-dependent operator on H and . We establish stability estimates of Hölder type and propose a regularization method with error estimates of Hölder type for the ill-posed backward semi-linear parabolic equation with the source function f satisfying a local Lipschitz condition.

On the adjoint of Hilbert space operators

In general, it is a non-trivial task to determine the adjoint of an unbounded operator S acting between two Hilbert spaces. We provide necessary and sufficient conditions for a given operator T to be identical with . In our considerations, a central role is played by the operator matrix . Our approach has several consequences such as characterizations of closed, normal, skew- and selfadjoint, unitary and orthogonal projection operators in real or complex Hilbert spaces. We also give a self-contained proof of the fact that always has a positive selfadjoint extension.

Scalar field dynamics in a BTZ background with generic boundary conditions

We revisit the dynamics of a massive scalar field in a Banados, Teitelboim, and Zanelli background taking into account the lack of global hyperbolicity of the spacetime. We approach this issue using the strategy of Ishibashi and Wald which finds a unique smooth solution as the causal evolution of initial data, each possible evolution corresponding to a positive self-adjoint extension of certain operator in a Hilbert space on the initial surface. Moreover, solutions obtained this way are the most general ones satisfying a few physically sensible requirements. This procedure is intimately related to the choice of boundary conditions and the existence of bound states. We find that the scalar field dynamics in the (effective) mass window -3/4le m_e^2ell ^2<0 can be well defined within a one-parametric family of distinct boundary conditions (-3/4 being the conformally coupled case), while for m_e^2ell ^2ge 0 the boundary condition is unique (only one self-adjoint extension is possible). It is argued that there is no sensible evolution possible for m_e^2ell ^2<-1, and also it is shown that in the range m_e^2ell ^2 in [-1,-3/4) there is a U(1) family of allowed boundary conditions, however, the positivity of the self-adjoint extensions is only motivated but not proven. We focus mainly on describing the dynamics of such evolutions given the initial data and all possible boundary conditions, and in particular we show the energy is always positive and conserved.

On the square root of a positive selfadjoint operator

We provide a short, elementary proof of the existence and uniqueness of the square root in the context of unbounded positive selfadjoint operators on real or complex Hilbert spaces.

The lack of exponential stability for a class of second-order systems with memory

We analyse the decay properties of the solution semigroup S(t) generated by the linear integrodifferential equationwhere the operator A is strictly positive self-adjoint with A–1 not necessarily compact. The asymptotic stability of S(t) is investigated in terms of the dependence of the parameter γ ∈ ℝ. In particular, we show that S(t) is not exponentially stable when γ ≠ 1.

Regularity for evolution equations with nonlocal initial conditions

This paper is concerned with the existence and uniqueness of global strong solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. We assume that the linear part is a positive definite selfadjoint operator, and the nonlinear part satisfies some essential growth conditions. The discussions are based on analytic semigroups theory, the piecewise regular method and relevant fixed point theorem. The results obtained in this paper improve and extend some related conclusions on this topic. An example to illustrate the feasibility of our abstract results is also given.

Symmetric Self-Adjoint Hopf Categories and a Categorical Heisenberg Double

We define what we call a symmetric self-adjoint Hopf structure on a semisimple abelian category, which is an analog of Zelevinsky’s positive self-adjoint Hopf algebra structure for categories. As examples we exhibit this structure on the categories of polynomial functors and equivariant polynomial functors and obtain a categorical manifestation of Zelevinsky’s decomposition theorem involving them. It follows from the work of Zelevinsky that every positive self-adjoint Hopf algebra A admits a Fock space action of the Heisenberg double (A,A). We show that the notion of symmetric self-adjoint Hopf category leads naturally to the definition of a categorical analog of such an action and that every symmetric self-adjoint Hopf category admits such an action.

Regularization of an inverse nonlinear parabolic problem with time-dependent coefficient and locally Lipschitz source term

We consider a backward problem of finding a function u satisfying a nonlinear parabolic equation in the form ut+a(t)Au(t)=f(t,u(t)) subject to the final condition u(T)=φ. Here A is a positive self-adjoint unbounded operator in a Hilbert space H and f satisfies a locally Lipschitz condition. This problem is ill-posed. Using quasi-reversibility method, we shall construct a regularized solution uε from the measured data aε and φε. We show that the regularized problems are well-posed and that their solutions converge to the exact solutions. Error estimates of logarithmic type are given and a simple numerical example is presented to illustrate the method as well as verify the error estimates given in the theoretical parts.

Gauge-Invariant Quasi-Free States on the Algebra of the Anyon Commutation Relations

Let $X=\mathbb R^2$ and let $q\in\mathbb C$, $|q|=1$. For $x=(x^1,x^2)$ and $y=(y^1,y^2)$ from $X^2$, we define a function $Q(x,y)$ to be equal to $q$ if $x^1<y^1$, to $\bar q$ if $x^1>y^1$, and to $\Re q$ if $x^1=y^1$. Let $\partial_x^+$, $\partial_x^-$ ($x\in X$) be operator-valued distributions such that $\partial_x^+$ is the adjoint of $\partial_x^-$. We say that $\partial_x^+$, $\partial_x^-$ satisfy the anyon commutation relations (ACR) if $\partial^+_x\partial_y^+=Q(y,x)\partial_y^+\partial_x^+$ for $x\ne y$ and $\partial^-_x\partial_y^+=\delta(x-y)+Q(x,y)\partial_y^+\partial^-_x$ for $(x,y)\in X^2$. In particular, for $q=1$, the ACR become the canonical commutation relations and for $q=-1$, the ACR become the canonical anticommutation relations. We define the ACR algebra as the algebra generated by operator-valued integrals of $\partial_x^+$, $\partial_x^-$. We construct a class of gauge-invariant quasi-free states on the ACR algebra. Each state from this class is completely determined by a positive self-adjoint operator $T$ on the real space $L^2(X,dx)$ which commutes with any operator of multiplication by a bounded function $\psi(x^1)$. In the case $\Re q<0$, the operator $T$ additionally satisfies $0\le T\le -1/\Re q$. Further, for $T=\kappa^2\mathbf 1$ ($\kappa>0$), we discuss the corresponding particle density $\rho(x):=\partial_x^+\partial_x^-$. For $\Re q\in(0,1]$, using a renormalization, we rigorously define a vacuum state on the commutative algebra generated by operator-valued integrals of $\rho(x)$. This state is given by a negative binomial point process. A scaling limit of these states as $\kappa\to\infty$ gives the gamma random measure, depending on parameter $\Re q$.

General theory of regular biorthogonal pairs and its physical operators

In this paper, we introduce a general theory of regular biorthogonal sequences and its physical operators. Biorthogonal sequences {ϕn} and {ψn} in a Hilbert space H are said to be regular if Span {ϕn} and Span {ψn} are dense in H. The first purpose is to show that there exists a non-singular positive self-adjoint operator Tf in H defined by an orthonormal basis (ONB) f ≡ {fn} in H such that ϕn = Tffn and ψn=Tf−1fn, n = 0, 1, …, and such an ONB f is unique. The second purpose is to define and study the lowering operators Af and Bf†, the raising operators Bf and Af†, and the number operators Nf and Nf† determined by the non-singular positive self-adjoint operator Tf. These operators connect with quasi-Hermitian quantum mechanics and its relatives. This paper clarifies and simplifies the mathematical structure of this framework and minimizes the required assumptions.

Irreducibility and Exponential Mixing of Some Stochastic Hydrodynamical Systems Driven by Pure Jump Noise

In this paper, we derive several results related to the long-time behavior of a class of stochastic semilinear evolution equations in a separable Hilbert space H: $$d\mathfrak{u}(t) +[{\rm A}\mathfrak{u}(t)+{\rm B}(\mathfrak{u}(t),\mathfrak{u}(t))] dt = dL(t), \quad\mathfrak{u}(0)=x\in {\rm H}.$$ Here A is a positive self-adjoint operator and B is a bilinear map, and the driving noise L is basically a \({D({\rm A}^{-1/2})}\) -valued levy process satisfying several technical assumptions. By using a density transformation theorem type for levy measure, we first prove a support theorem and an irreducibility property of the Ornstein–Uhlenbeck processes associated to the nonlinear stochastic problem. Second, by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain \({\gamma \in [0,1/4]}\)B is continuous on \({D({\rm A}^\gamma)\times D({\rm A}^\gamma)}\) with values in \({D({\rm A}^ {-1/2})}\). Using a coupling argument, the exponential ergodicity is also proved under the stronger assumption that B is continuous on \({{\rm H} \times{\rm H}}\). While the latter condition is only satisfied by the nonlinearities of GOY and Sabra shell models, the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2D Navier–Stokes, Magnetohydrodynamics equations, the 3D Leray-\({{\varvec{\alpha}}}\) model, the GOY and Sabra shell models.

The Moore–Gibson–Thompson equation with memory in the critical case

We consider the following abstract version of the Moore–Gibson–Thompson equation with memory∂tttu(t)+α∂ttu(t)+βA∂tu(t)+γAu(t)−∫0tg(s)Au(t−s)ds=0 depending on the parameters α,β,γ>0, where A is strictly positive selfadjoint linear operator and g is a convex (nonnegative) memory kernel. In the subcritical case αβ>γ, the related energy has been shown to decay exponentially in [19]. Here we discuss the critical case αβ=γ, and we prove that exponential stability occurs if and only if A is a bounded operator. Nonetheless, the energy decays to zero when A is unbounded as well.

Hopf Modules and Representations of Finite Wreath Products

For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower $G \wr S_{n} := S_{n} \ltimes G^{n}$ of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K 0(R e p−G≀S n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ⊂ G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k t h -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).

$$L_1$$ L 1 -space for a positive operator affiliated with von Neumann algebra

In this paper we suggest an approach for constructing an L1-type space for a positive selfadjoint operator affiliated with von Neumann algebra. For such operator we intro- duce a seminorm, and prove that it is a norm if and only if the operator is injective. For this norm we construct an L1 -type space as the complition of the space of hermitian ultraweakly continuous linear functionals on von Neumann algebra, and represent L1- type space as a space of continuous linear functionals on the space of special sesquilinear forms. Also, we prove that L1 -type space is isometrically isomorphic to the predual of von Neumann algebra in a natural way. We give a small list of alternate definitions of the seminorm, and a special definition for the case of semifinite von Neumann algebra, in particular. We study order properties of L1-type space, and demonstrate the con- nection between semifinite normal weights and positive elements of this space. At last, we construct a similar L-space for the positive element of C*-algebra, and study the connection between this L-space and the L1 -type space in case when this C*-algebra is a von Neumann algebra.