Positive Self-adjoint Operator Research Articles

Some remarks on quasi-Hermitian operators

A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert C*-module l 2(A)

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module \(l_2 \bar \otimes A\), and give a one to one correspondence between the set of positive self-adjoint regular module operators on \(l_2 \bar \otimes A\) and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup \(\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)\) is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L(\(l_2 \bar \otimes A\)) is the von Neumann algebra of all adjointable module maps on \(l_2 \bar \otimes A\).

A sufficient condition for a discrete spectrum of the Kirchhoff plate with an infinite peak

Elliptic boundary value problems on domains which have a Lipschitz boundary and a compact closure, in particular when they generate positive self-adjoint operators, have fully discrete spectra. However, if the domain loses the Lipschitz property or compactness, other situations may occur. It is well-known that for the Dirichlet case boundedness is sufficient but not necessary for a discrete spectrum. See the famous paper by Rellich [26] or the more recent contributions by [27, 3]. On the other hand, for the Neumann problem of the Laplace operator there exist numerous examples of bounded domains such that the spectrum gets a non-empty continuous component (see e.g. [7, 17, 18, 28, 11]). The literature on the spectra for the Laplace operator with various boundary conditions is focussed on domains that have a cusp, a finite or infinite peak or horn [3, 11, 13, 8, 14, 12, 6, 4, 15] or even considered a rolled horn [28]. The criteria in [1] and [9] for the embedding H1(Ω) ⊂ L2(Ω) to be compact, show that the Neumann-Laplace problem on a domain Ω with the infinite peak

Sobolev-like cones of trace-class operators on unbounded domains: Interpolation inequalities and compactness properties

In this paper we extend the compactness properties for trace-class operators obtained by Dolbeault, Felmer and Mayorga-Zambrano to a smooth unbounded domain Ω⊆Rd, d≥3. We consider V, a non-negative potential on Ω that blows up at infinity, and the normed space HV(Ω)={u∈H01(Ω):‖u‖V2=∫Ω(∣∇u(x)∣2+∣u(x)∣2V(x))dx<∞}. A positive self-adjoint trace-class operator R belongs to the Sobolev-like cone HV,+1 if (ψi,R)N⊆HV(Ω) and 《R》V=∑i=1∞νi,R‖ψi,R‖V2<∞, where (νi,R)i∈N is the sequence of occupation numbers of R and (ψi,R)i∈N⊆L2(Ω) is a corresponding Hilbertian basis of eigenfunctions. We prove that a sequence in HV,+1, bounded in energy 《⋅》V, has a subsequence that converges in trace norm; this is analogous to the classical Sobolev immersion H1(Ω)⊆L2(Ω). We prove the existence of lower bounds for nonlinear free energy functionals and, by doing so, we establish Lieb–Thirring type inequalities as well as some Gagliardo–Nirenberg type interpolation inequalities; then our compactness result is applied to minimize nonlinear free energy functionals working on HV,+1.

Compactness of trajectories to some nonlinear second order evolution equations and applications

Under suitable growth and coercivity conditions on the nonlinear damping operator g, we establish boundedness or compactness properties of trajectories to the equationu¨(t)+g(u˙(t))+Au(t)=h(t),t∈R+, where A is a positive selfadjoint operator. The compactness results are used to prove the existence of almost periodic solutions when h is almost periodic, and to generalize some recent results of Chergui and Ben Hassen–Chergui concerning convergence to equilibrium when a nonlinear term depending on u is added and h dies off sufficiently fast for t large.

The characterization of a class of quantum Markov semigroups and the associated operator-valued Dirichlet forms based on Hilbert W*-module

In this paper, we introduce the concept of operator-valued quadratic form based on Hilbert W*-module $l_2 \bar \otimes A$ , and give a one to one correspondence between the set of positive self-adjoint regular module operators on $l_2 \bar \otimes A$ and the set of regular quadratic forms, where A is a finite and σ-finite von Neumann algebra. Furthermore, we obtain that a strict continuous symmetric regular module operator semigroup $\{ T_t \} _{t \in \mathbb{R}^ + } \subset L(l_2 \bar \otimes A)$ is Markovian if and only if the associated A-valued quadratic form is a Dirichlet form, where L( $l_2 \bar \otimes A$ ) is the von Neumann algebra of all adjointable module maps on $l_2 \bar \otimes A$ .

On form sums of positive operators

The purpose of the present note is to provide domain, kernel and range characterizations for the form sum of two positive selfadjoint operators. In addition, we establish a criterion for the closedness of the range of the form sum and give the Moore–Penrose pseudoinverse in this case.

Analyticity and regularity for a class of second order evolution equations

The regularity conservation as well as the smoothing effect are studied for the equation$ u''+ Au+ cA^\alpha u' = 0$, where $A$ is a positive selfadjoint operator on a real Hilbert space $H$ and$\alpha\in (0, 1]; \,\, c >0$. When $\alpha\ge {1\over 2}$ the equation generates an analytic semigroup on$D(A^{1/2})\times H $ , and if $\alpha\in (0, {1\over 2})$ a weaker optimal smoothing property is established.Some conservation properties in other norms are also established, as a typical example, the strongly dissipative wave equation$u_{tt} - \Delta u -c\Delta u_t = 0$ with Dirichlet boundary conditions in a bounded domainis given, for which the space$C_0(\Omega)\times C_0(\Omega)$ is conserved for $t>0$, which presents a sharp contrast with the conservative case$u_{tt} - \Delta u = 0$ for which $C_0(\Omega)$-regularity can be lost even starting from an initial state$(u_0, 0)$ with $u_0\in C_0(\Omega)\cap C^1(\overline {\Omega})$.

Spectral Regularization Methods for an Abstract Ill-Posed Elliptic Problem

We study an abstract elliptic Cauchy problem associated with an unbounded self-adjoint positive operator which has a continuous spectrum. It is well-known that such a problem is severely ill-posed; that is, the solution does not depend continuously on the Cauchy data. We propose two spectral regularization methods to construct an approximate stable solution to our original problem. Finally, some other convergence results including some explicit convergence rates are also established under a priori bound assumptions on the exact solution.

Asymptotic equipartition of operator-weighted energies in damped wave equations

We prove an asymptotic energy equipartition result for abstract damped wave equations of the form utt + 2F (S)ut + S 2 u = 0, where S is a strictly positive self-adjoint operator and the damping operator F (S) is small. This means that under certain assumptions, the ratio of suitably modified kinetic and potential energies, K(t)/ P (t), tends to 1 as t →∞ for all nonzero solutions u(t) of the equation. Here, K(t )a nd P (t) are conveniently weighted versions of the usual kinetic and potential energies of the associated undamped equation. Previous results, concerning the undamped case and the scalar-damped one, are particular cases. We propose an extension of the concepts of hyperbolicity and unitarity that allows one to consider the equipartition property in a more general setting. Some examples involving PDEs, as well as pseudo-differential equations, are given.

Stability estimates for a class of semi-linear ill-posed problems

We study a final value problem for a nonlinear parabolic equation with positive self-adjoint unbounded operator coefficients. The problem is ill-posed. The regularized equation is given by a modified quasi-reversibility method. For this regularization solution, the Hölder type stability estimate between the regularization solution and the exact solution is obtained.

Resolvent conditions for the control of parabolic equations

Since the seminal work of Russell and Weiss in 1994, resolvent conditions for various notions of admissibility, observability and controllability, and for various notions of linear evolution equations have been investigated intensively, sometimes under the name of infinite-dimensional Hautus test. This paper sets out resolvent conditions for null-controllability in arbitrary time: necessary for general semigroups, sufficient for analytic normal semigroups. For a positive self-adjoint operator A, it gives a sufficient condition for the null-controllability of the semigroup generated by −A which is only logarithmically stronger than the usual condition for the unitary group generated by iA. This condition is sharp when the observation operator is bounded. The proof combines the so-called “control transmutation method” and a new version of the “direct Lebeau–Robbiano strategy”. The improvement of this strategy also yields interior null-controllability of new logarithmic anomalous diffusions.

A local version of Hardy spaces associated with operators on metric spaces

Let (X, d, µ) be a metric measure space endowed with a distance d and a nonnegative Borel doubling measure µ. Let L be a second order self-adjoint positive operator on L 2(X). Assume that the semigroup e −tL generated by −L satisfies the Gaussian upper bounds on L 2(X). In this article we study a local version of Hardy space h 1 (X) associated with L in terms of the area function characterization, and prove their atomic characters. Furthermore, we introduce a Moser type local boundedness condition for L, and then we apply this condition to show that the space h 1 (X) can be characterized in terms of the Littlewood-Paley function. Finally, a broad class of applications of these results is described.

On weighted monotonicity inequalities and characterizations of the traces

Let τ be a faithful normal semi-finite trace on a von Neumann algebra M, \(\tilde \tau \) is the extension of τ on the set of all (in general, unbounded) self-adjoint positive operators affiliated with M. We study inequalities of the form $$\tilde \tau (w(A)^{1/2} f(A)w(A)^{1/2} ) \leqslant \tilde \tau (w(A)^{1/2} f(B)w(A)^{1/2} ),$$ where f, w are nonnegative real-valued functions, A and B are unbounded self-adjoint positive operators affiliated with algebra M such that A ≤ B. A new characterization of the traces related to these inequalities will be shown.

Weyl asymptotics of bisingular operators and Dirichlet divisor problem

We consider a class of pseudodifferential operators, with crossed vector valued symbols, defined on the product of two closed manifolds. We study the asymptotic expansion of the counting function of positive selfadjoint operators in this class. Using a general Theorem of Aramaki, we can determine the first term of the asymptotic expansion of the counting function and, in a special case, we are able to find the second term. We give also some examples, emphasizing connections with problems of analytic number theory, in particular with Dirichlet divisor function.

G-Convergence of Dirac Operators

We consider the linear Dirac operator with a (<svg style="vertical-align:-0.0pt;width:18.65px;" id="M1" height="10.6875" version="1.1" viewBox="0 0 18.65 10.6875" width="18.65" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(1.25,0,0,-1.25,0,10.6875)"> <g transform="translate(72,-63.45)"> <text transform="matrix(1,0,0,-1,-71.95,63.5)"> <tspan style="font-size: 12.50px; " x="0" y="0">−</tspan> <tspan style="font-size: 12.50px; " x="8.5645552" y="0">1</tspan> </text> </g> </g> </svg>)-homogeneous locally periodic potential that varies with respect to a small parameter. Using the notation of G-convergence for positive self-adjoint operators in Hilbert spaces we prove G-compactness in the strong resolvent sense for families of projections of Dirac operators. We also prove convergence of the corresponding point spectrum in the spectral gap.

Operator extensions with closed range

We characterize those positive (not necessarily densely defined) operators whose Krein-von Neumann extension, the smallest among all positive selfadjoint extensions, has closed range. In addition, we construct their Moore- Penrose pseudoinverse by employing factorization via an auxiliary Hilbert space. Other extremal extensions, in particular the Friedrichs extension, are also in- vestigated from this point of view. As an application, new characterizations of essentially selfadjoint positive operators are presented.

On the stability of the massive scalar field in Kerr space-time

The current early stage in the investigation of the stability of the Kerr metric is characterized by the study of appropriate model problems. Particularly interesting is the problem of the stability of the solutions of the Klein-Gordon equation, describing the propagation of a scalar field in the background of a rotating (Kerr-) black hole. Results suggest that the stability of the field depends crucially on its mass μ. Among others, the paper provides an improved bound for μ above which the solutions of the reduced, by separation in the azimuth angle in Boyer-Lindquist coordinates, Klein-Gordon equation are stable. Finally, it gives new formulations of the reduced equation, in particular, in form of a time-dependent wave equation that is governed by a family of unitarily equivalent positive self-adjoint operators. The latter formulation might turn out useful for further investigation. On the other hand, it is proved that from the abstract properties of this family alone it cannot be concluded that the corresponding solutions are stable.

Proof of a Modified Jaynes's Estimation Theory

It is proved that the state of maximum entropy, having observed values for the n observables, X1,…,Xn, is the same state that minimises the matrix of covariances of any n locally unbiased estimators for n parameters for the probability distribution of X1,…,Xn. We sketch how to get a similar result in quantum theory, in which X1,…,Xn are (not necessarily commuting) quadratic forms that are bounded relative to a positive self-adjoint operator H such that exp (-βH) is of trace-class for some positive β.

Existence and asymptotic stability of periodic solution for evolution equations with delays

In this paper, we discuss the existence and asymptotic stability of the time periodic solution for the evolution equation with multiple delays in a Hilbert space H u ′ ( t ) + A u ( t ) = F ( t , u ( t ) , u ( t − τ 1 ) , … , u ( t − τ n ) ) , t ∈ R , where A : D ( A ) ⊂ H → H is a positive definite selfadjoint operator, F : R × H n + 1 → H is a nonlinear mapping which is ω-periodic in t, and τ 1 , τ 2 , … , τ n are positive constants. We present essential conditions on the nonlinearity F to guarantee that the equation has ω-periodic solutions or an asymptotically stable ω-periodic solution. The discussion is based on analytic semigroups theory and an integral inequality with delays.