Certain Class Of Operators Research Articles

Maximal numerical ranges of certain classes of operators and approximation

Maximal numerical ranges of certain classes of operators and approximation

Time-frequency analysis and coorbit spaces of operators

To analyse the time-frequency content of an ordered data set, it is desirable to consider cross-correlation of time-frequency contents of data points, while maintaining the order of such correlations. To this end, we introduce an operator-valued short-time Fourier transform for certain classes of operators with operator windows, and show that the transform acts in a way analogous to the way the short-time Fourier transform for functions acts, in particular giving rise to a family of vector-valued reproducing kernel Banach spaces, the so-called coorbit spaces, as spaces of operators. As a result of this structure, the operators generating equivalent norms on the function modulation spaces are fully classified. We show that these classes of operators have the same atomic decomposition properties as the function spaces, and use this to give a characterisation of the spaces using localisation operators.

Numerical radius and Berezin number inequality

We study various inequalities for numerical radius and Berezin number of a bounded linear operator on a Hilbert space. It is proved that the numerical radius of a pure two-isometry is 1 and the Crawford number of a pure two-isometry is 0. In particular, we show that for any scalar-valued non-constant inner function θ, the numerical radius and the Crawford number of a Toeplitz operator Tθ on a Hardy space is 1 and 0, respectively. It is also shown that numerical radius is multiplicative for a class of isometries and sub-multiplicative for a class of commutants of a shift. We have illustrated these results with some concrete examples. Finally, some Hardy-type inequalities for Berezin number of certain class of operators are established with the help of the classical Hardy's inequality.

Smooth Perturbations of the Functional Calculus and Applications to Riemannian Geometry on Spaces of Metrics

We show for a certain class of operators A and holomorphic functions f that the functional calculus $$A\mapsto f(A)$$ is holomorphic. Using this result we are able to prove that fractional Laplacians $$(1+\Delta ^g)^p$$ depend real analytically on the metric g in suitable Sobolev topologies. As an application we obtain local well-posedness of the geodesic equation for fractional Sobolev metrics on the space of all Riemannian metrics.

On the p-numerical radii of Hilbert space operators

In this paper, we give new results for the p-numerical radii of Hilbert space operators. It is shown, among other inequalities, that if A is a Hilbert space operator, which belongs to the Schatten p-class, then and for . Also, and for , where is the Schatten p-norm. Applications of these inequalities to certain classes of operators are also given.

Certain Classes of Operators on Some Weighted Hyperbolic Function Spaces

In this paper, some classes of concerned multiplication operators consisting of analytic and hyperbolic functions are defined and considered. Furthermore, some properties such as boundedness and compactness of the new operators are discussed. Finally, a general class of weighted hyperbolic Bloch functions is characterized by metric spaces.

Quantum uncertainty as classical uncertainty of real-deterministic variables constructed from complex weak values and a global random variable

What does it take for real-deterministic $c$-valued (i.e., classical, commuting) variables to comply with the Heisenberg uncertainty principle? Here, we construct a class of real-deterministic $c$-valued variables out of the weak values obtained via a nonperturbing weak measurement of quantum operators with a postselection over a complete set of state vectors basis, which always satisfies the Kennard-Robertson-Schr\"odinger uncertainty relation. First, we introduce an auxiliary global random variable and couple it to the imaginary part of the weak value to convert the incompatibility between the quantum operator and the basis into the fluctuation of an ``error term,'' and then superimpose it onto the real part of the weak value. We show that this class of ``$c$-valued physical quantities'' provides a real-deterministic contextual hidden-variable model for the quantum expectation value of a certain class of operators. We then show that the Schr\"odinger and the Kennard-Robertson lower bounds can be obtained separately by imposing the classical uncertainty relation to the $c$-valued physical quantities associated with a pair of Hermitian operators. Within the representation, the complementarity between two incompatible quantum observables manifests the absence of a basis wherein the error terms of the associated two $c$-valued physical quantities simultaneously vanish. Furthermore, quantum uncertainty principle is captured by a specific irreducible epistemic restriction between the two $c$-valued physical quantities, foreign in classical mechanics, constraining the allowed form of their joint distribution. We then suggest an epistemic interpretation of the two terms decomposing the $c$-valued physical quantity as the optimal estimate under the epistemic restriction and the associated estimation error, and discuss the classical limit.

On spin structures and orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of connections on P modulo gauge, as a topological stack. For a real elliptic operator E• we previously studied orientations on the real determinant line bundle over BP, twisting E• by connections ∇Ad(P). These are used to construct orientations in the usual sense on smooth gauge theory moduli spaces, and have been extensively studied since the work of Donaldson [15–17].Here we consider complex elliptic operators F• and introduce the idea of spin structures, square roots of the complex determinant line bundle of F• twisted over BP. These may be used to construct spin structures in the usual sense on smooth complex gauge theory moduli spaces. We study the existence and classification of such spin structures.Our main result identifies spin structures on X with orientations on X×S1. Thus, if P→X and Q→X×S1 are principal G-bundles with Q|X×{1}≅P, we relate spin structures on (BP,F•) to orientations on (BQ,E•) for a certain class of operators F• on X and E• on X×S1.Combined with [25], we obtain canonical spin structures for positive Diracians on spin 6-manifolds and gauge groups G=U(m),SU(m). In a sequel [26] we will apply this to define canonical orientation data for all Calabi–Yau 3-folds X over C, as in Kontsevich and Soibelman [28, §5.2], solving a long-standing problem in Donaldson–Thomas theory.

Constants of strong uniqueness of minimal projections onto some n-dimensional subspaces of [formula omitted

In this paper, we find strong uniqueness constants for a certain class of operators with a unit norm from a space of dimension 2n onto its subspace of codimension n, which is formed by using hyperplanes in l∞2n(n≥2).

Joint discrete universality for periodic zeta-functions. II

In the paper, for certain classes of operators F in the space of analytic functions, we prove the discrete universality for compositions F (ζ(s, α ; 𝔞, 𝔟 )), where ζ(s, α ; 𝔞 ; 𝔟 ) is a collection consisting from periodic and periodic Hurwitz zeta-functions, i. e., the approximation of analytic functions by discrete shifts F (ζ(s + ikh, α ; 𝔞 ; 𝔟 )) with h > 0 and k = 0, 1, . . For this, a theorem of [12] is applied.

An Application of Infinite Sums and Products Relating to Spectral Synthesis

The following is a discussion regarding a specific set of operators acting on the space of functions analytic on the unit disk. A diagonal operator is said to admit spectral synthesis if all of its invariant subspaces can be expressed as a closed linear span of a subset of its eigenvectors. This article employs various techniques for verifying the convergence of infinite products and infinite sums as a means of demonstrating that a certain class of operators fail to admit spectral synthesis.

Function Weighted Quasi-Metric Spaces and Fixed Point Results

Hereafter, the concept of a function weighted quasi-metric space is introduced. A necessary topology on this new structure is considered. A condition that ensures the uniqueness of the limit of a sequence in such a space is provided. A relation between the bi-convergent sequences and the bi-Cauchy ones are proved. Certain classes of operators with respect to their fixed point properties are investigated, having in view this framework. Examples that support our results are also included.

Instantons via breaking geometric symmetry in hyperbolic traps

Using geometrical and algebraic ideas, we study tunnel eigenvalue asymptotics and tunnel bilocalization of eigenstates for certain class of operators (quantum Hamiltonians) including the case of Penning traps, well known in physical literature. For general hyperbolic traps with geometric asymmetry, we study resonance regimes which produce hyperbolic type algebras of integrals of motion. Such algebras have polynomial (non-Lie) commutation relations with creation-annihilation structure. Over this algebra, the trap asymmetry (higher-order anharmonic terms near the equilibrium) determines a pendulum-like Hamiltonian in action-angle coordinates. The symmetry breaking term generates a tunneling pseudoparticle (closed instanton). We study the instanton action and the corresponding spectral splitting.

Bogolyubov inequality for the ground state and its application to interacting rotor systems

We have formulated and proved the Bogolyubov inequality for operators at zero temperature. So far this inequality has been known for matrices, and we were able to extend it to certain class of operators. We have also applied this inequality to the system of interacting rotors. We have shown that if: {\em i)} the dimension of the lattice is 1 or 2, {\em ii)} the interaction decreases sufficiently fast with a distance, and {\em iii)} there is an energy gap over the ground state, then the spontaneous magnetization in the ground state is zero, i.e. there is no LRO in the system. We present also heuristic arguments (of perturbation-theoretic nature) suggesting that one- and two-dimensional system of interacting rotors has the energy gap independent of the system size if the interaction is sufficiently small.

Universal mappings for certain classes of operators and polynomials between Banach spaces

AbstractA well‐known result of J. Lindenstrauss and A. Pełczyński (1968) gives the existence of a universal non‐weakly compact operator between Banach spaces. We show the existence of universal non‐Rosenthal, non‐limited, and non‐Grothendieck operators. We also prove that there does not exist a universal non‐Dunford–Pettis operator, but there is a universal class of non‐Dunford–Pettis operators. Moreover, we show that, for several classes of polynomials between Banach spaces, including the non‐weakly compact polynomials, there does not exist a universal polynomial.

On the Uniqueness of Jordan Canonical Form Decompositions of Operators by K-theoretical Data

AbstractIn this paper, we develop a generalized Jordan canonical form theorem for a certain class of operators in ℒ(Η). A complete criterion for similarity for this class of operators in terms of K-theory for Banach algebras is given.

Applications of Fourier Analysis in Homogenization of the Dirichlet Problem: L p Estimates

Let u ɛ be a solution to the system $$\rm div(A_\varepsilon(x)\nabla u_\varepsilon(x)) = 0 \quad\text{in}\, D,\quad u_\varepsilon(x) = g(x,x/\varepsilon)\quad\text{on} \,\partial\, D,$$ where $${D \subset \mathbb{R}^d (d \geqq 2)}$$ , is a smooth uniformly convex domain, and g is 1-periodic in its second variable, and both A ɛ and g are sufficiently smooth. Our results in this paper are twofold. First we prove L p convergence results for solutions of the above system and for the non oscillating operator $${A_\varepsilon(x) = A(x)}$$ , with the following convergence rate for all $${1\leqq p < \infty}$$ $$\left.\begin{array}{ll}\parallel\,u_\varepsilon - u_0\parallel{L^p(D)} \leqq C_p \left\{\begin{array}{ll}\varepsilon^{1/2p}, \quad\quad\quad\quad d=2, \\(\varepsilon \mid {\rm ln} \varepsilon \mid)^{1/p}, \quad d = 3, \\ \varepsilon^{1/p}, \quad\quad\quad\quad d \geqq 4,\end{array}\right.\end{array}\right.$$ which we prove is (generically) sharp for $${d \geqq 4}$$ . Here u 0 is the solution to the averaging problem. Second, combining our method with the recent results due to Kenig, Lin and Shen (Commun Pure Appl Math 67(8):1219–1262, 2014), we prove (for certain class of operators and when $${d \geqq 3}$$ ) $$\parallel\,u_\varepsilon - u_0\parallel\,L^p(D) \leqq C_p [\varepsilon ({\rm ln}(1/ \varepsilon))^2 ]^{1/p}$$ for both the oscillating operator and boundary data. For this case, we take $${A_\varepsilon = A(x/ \varepsilon)}$$ , where A is 1-periodic as well. Some further applications of the method to the homogenization of the Neumann problem with oscillating boundary data are also considered.

The Gaussian-smoothed Wigner function and its application to precision analysis

We study a class of phase-space distribution functions that is generated from a Gaussian convolution of the Wigner distribution function. This class of functions represents the joint count probability in simultaneous measurements of position and momentum. We show that, using these functions, one can determine the expectation value of a certain class of operators accurately, even if measurement data performed only with imperfect detectors are available. As an illustration, we consider the eight-port homodyne detection experiment that performs simultaneous measurements of two quadrature amplitudes of a radiation field.

Two‐Phase Free Boundary Problems for Parabolic Operators: Smoothness of the Front

We continue to develop the regularity theory of general two‐phase free boundary problems for parabolic operators. In a 2010 paper, we establish the optimal (Lipschitz) regularity of a viscosity solution under the assumptions that the free boundary is locally a flat Lipschitz graph and a nondegeneracy condition holds. Here, on one side we improve this result by removing the nondegeneracy assumption; on the other side we prove the smoothness of the front. The proof relies in a crucial way on a local stability result stating that, for a certain class of operators, under small perturbations of the coefficients flat free boundaries remain close and flat. © 2013 Wiley Periodicals, Inc.

Fixed-Point Methods for a Certain Class of Operators

We introduce in this paper a new class of nonlinear operators, which contains, among others, the class of operators with semimonotone additive inverse and also the class of nonexpansive mappings. We study this class and discuss some of its properties. Then we present iterative procedures for computing fixed points of operators in this class, which allow for inexact solutions of the subproblems and relative error criteria. We prove weak convergence of the generated sequences in the context of Hilbert spaces. Strong convergence is also discussed.