A VIEWPOINT TO MEASURE OF NON-COMPACTNESS OF OPERATORS IN BANACH SPACES

This article is committed to deal with measure of non-compactness of operators in Banach spaces.Firstly,the collection C(X)(consisting of all nonempty closed bounded convex sets of a Banach space X endowed with the uaual set addition and scaler multiplication)is a normed semigroup,and the mapping J from C(X)onto F(Ω)is a fully order-preserving positively linear surjective isometry,whereΩis the closed unit ball of X^*and F(Ω)the collection of all continuous and w^*-lower semicontinuous sublinear functions on X^*but restricted toΩ.Furthermore,both EC=JC-JC and EK=JK-JK are Banach lattices and EK is a lattice ideal of EC.The quotient space EC/EK is an abstract M space,hence,order isometric to a sublattice of C(K)for some compact Haudorspace K,and(FQJ)C which is a closed cone is contained in the positive cone of C(K),where Q:EC→EC/EK is the quotient mapping and F:EC/EK→C(K)is a corresponding order isometry.Finally,the representation of the measure of non-compactness of operators is given:Let BX be the closed unit ball of a Banach space X,thenμ(T)=μ(T(BX))=||(F QJ)T(BX)||C(K),∀T∈B(X).

Remote interactions on two distributed quantum systems:nonlocal unambiguous quantum-state discrimination

Remote quantum-state discrimination is a critical step for the implementation of quantum communication network and distributed quantum computation. We present a protocol for remotely implementing the unambiguous discrimination between nonorthogonal states using quantum entanglements, local operations, and classical communications. This protocol consists of a remote generalized measurement described by a positive operator valued measurement （POVM）. We explicitly construct the required remote POVM. The remote POVM can be realized by performing a nonlocal controlled-rotation operation on two spatially separated qubits, one is an ancillary qubit and the other is the qubit which is encoded by two nonorthogonal states to be distinguished, and a conventional local Von Neumann orthogonal measurement on the ancilla. The particular pair of states that can be remotely and unambiguously distinguished is specified by the state of the ancilla. The probability of successful discrimination is not optimal for all admissible pairs. However, for some subset it can be very close to an optimal value in an ordinary local POVM.

Teleporting a quantum controlled-Not with one target/two targets gate using two partially entangled states

This paper considers the teleportation of quantum controlled-Not （CNOT） gate by using partially entangled states. Different from the known probability schemes, it presents a method for teleporting a CNOT gate with unit fidelity and unit probability by using two partially entangled pairs as quantum channel. The method is applicable to any two partially entangled pairs satisfying the condition that their smaller Schmidt coefficients μ and ν are （2μ ＋ 2ν - 2μν - 1） ≥ 0. In this scheme, the sender＇s local generalized measurement described by a positive operator valued measurement （POVM） lies at the heart. It constructs the required POVM. It also puts forward a scheme for teleporting a CNOT with two targets gate with unit fidelity by using same quantum channel. With assistance of local operations and classical communications, three spatially separated users are able to complete the teleportation of a CNOT with two targets gate with probability of （2μ ＋ 2ν - 1）. With a proper value of μ and ν, the probability could reach nearly 1.

Dynamical time versus system time in quantum mechanics

Properties of an operator representing the dynamical time in the extended parameterization invariant formulation of quantum mechanics are studied. It is shown that this time operator is given by a positive operator measure analogously to the quantities that are known to represent various measurable time operators. The relation between the dynamical time of the extended formulation and the best known example of the system time operator, i.e., for the free one- dimensional particle, is obtained.

Phase Properties of Nonlinear Coherent States

Using the Pegg-Barnett formalism we study the phase probability distributions and the squeezing effects of measured phase operators in the nonlinear coherent states introduced by R.L. de Matos Filho and W. Vogel to describe the center-of mass motion of a trapped ion and the q-coherent states. Moreover, we have obtained the completeness relation of nonlinear coherent states and proved that the q-Fock state \n>(q) introduced in many papers is, in fact, the usual Fock state.

Fluctuations and Squeezing for Measured Phase Operators of the J-C Model in the Kerr Medium

By using the theory of measured phase operator proposed by Barnett and Pegg, dynamic properties of the phase of a field are studied. The time evolution and squeezing of measured phase operators of a coherent field interacting with a two-level atom in the cavity with or without the Kerr medium are investigated. The influences of virtual cavity field on squeezing of measured phase operator are studied. Our numerical results show that the squeezing effects are clearly influenced by Kerr medium parameters, the field intensity, and the detuning. Moreover, the influence of the virtual-photon field makes more quantum noise in the evolution of measured phase operators. Key words Jaynes-Cummings model (JCM) - Kerr medium - measured phase operator - squeezing - virtual photon PACS 2001 4250Dv

Higher-Order Fluctuations and Measured Phase Operators in the Squeezed Thermal States

We study the higher order fluctuations and squeezing of quadrature operators in the squeezed thermal states. In terms of measured phase operators, we discuss the fluctuations and squeezing of phases in these states. We conclude that the condition of higher order squeezing for quadrature components of the field is order independent and the fluctuations of measured phase operators are temperature independent.

TWO GENERALIZATIONS OF HARDY-LITTLEWOOD MAXIMAL OPERATOR

Two generalizations of Hardy-Littlewood maximal operator are considered. Some estimates for them are obtained.

BOUNDEDNESS OF VECTOR-VALUED CALDERN-ZYGMUND OPERATORS ON HERZ SPACES WITH NON-DOUBLING MEASURES

In the paper we obtain vector-valued inequalities for Calderon-Zygmund operator, simply CZO on Herz space and weak Herz space. In particular, we obtain vector-valued inequalities for CZO on L^q（R^d,｜x｜^α d μ）space, with 1〈q〈∞,-n〈α〈n（q-1）,and on L^1,∞ （R^d,｜x｜^α d μ）space,with -n〈α〈0.

Properties of Phase and Their Evolutions in Damped Odd and Even Coherent States

The properties of measured phase operators in damped odd and even coherent states have been studied. The fluctuations associated with measured phase and their squeezing in these states are investigated. The phase properties in damped superposition coherent states are considered too with the help of measured phase operators. These fluctuations and their squeezing are affected by damping and evolve with time elapsing.

Impact factors and improvement measures of enviroment quality in cleaning operating room

Objective To analyze the factors that impact the environment quality of cleaning operating room and to discuss improvement measurement. Methods Environment bacteria testing results and prevalence of surgical incision infection in cleaning operating room were retro-

Nonlocal unambiguous discrimination among N nonorthogonal qudit states lying in a higher-dimensional Hilbert space

We give a strategy for nonlocal unambiguous discrimination (UD) among N linearly independent nonorthogonal qudit states lying in a higher-dimensional Hilbert space. The procedure we use is a nonlocal positive operator valued measurement (POVM) in a direct sum space. This scheme is designed for obtaining the conclusive nonlocal measurement results with a finite probability of success. We construct a quantum network for realizing the nonlocal UD with a set of two-level remote rotations, and thus provide a feasible physical means to realize the nonlocal UD.

Laplace-Beltrami Operators of Measures on Banach Spaces

0. Introduction Let X be a real separable Banach space and X~* be its dual space, Let B(X) be the Borel field, i.e., the topological σ-field. A functional u: X→R’ is called a bounded smooth functional, if n∈N, f1, …, fn∈ X~* and φ∈Cb~∞(Rn), such that

Bounded and Compact Toeplitz Operators on the Weighted Bergman Space over the Bidisk

Forα〉1,let dvαdenote the weighted Lebesgue measure on the bidisk andμa complex measure satisfying some Carleson-type conditions.In this paper,we show a sufcient and necessary condition for the Toeplitz operatorTαˉμto be bounded or compact on weighted Bergman spaceL1a（dvα）.

Carleson measures, BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L^2(R^n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R^n) denote the BMO space associated to the Schrdinger operator L on R^n. In this article, we show that for every f ∈ BMO_L(R^n) with compact support, then there exist g ∈ L~∞(R^n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R^n), where S_(μ,P)=∫(R_+^(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-^(t(L)^(1/2))}t>0 on L^2(R^n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R^n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.