Operator Matrix Forms of Positive Operators

If a 3-tuple (A:H_1→H_1,B:H_2→H_1,C:H_2→H_2) of operators on Hilbert spaces is given,we proved that the operator A:= on H=H_1⊕H_2 is≥0 if and only if A≥0,R(B) R(A1/2)and C≥B~* A^+ B, where A^+ is the generalized inverse of A. In general,A^+ is a closed operator,but since R(B) R(A1/2),B~* A^+ B is bounded yet.

The Spectrum of Positive Operators on Hilbert Lattices

Some spectral characterizations of positive operators on Hilbert lattices are presented. The application of these results can yield some equivalent relations of an irreducible positive operator. Some related results for positive operators on Hilbert lattice are also obtained.

MEAN CONVERGENCE OF SOME POSITIVE OPERATORS

L'convergence with 0<p<infinity of the positive operators F.(da,f) and G.(da,f), introduced by Nevai P. [3], and Qn,8(da,f), introduced by Hermann T. and Vertesi P. [1] to f is an element of C[-1,1] is proved to hold for very general measures.

A New Class of Generalized Landau Linear Positive Operator Sequence and Its Properties of Approximation

In this paper two sequences of generalized Landau linear positive operators are introduced. They can be applied in approximating continuous functions with arbitrary growth order, defined on a finite interval or the whole real axis. The properties of approximation are studied and their asymptotic formulae are presented. These results show that their degrees of approximation are the best among existing operator sequences of Landau type, for example, their degrees of approximation for C 2[0, 1] are O(1/n 2) but corresponding degree of ordinary Landau operators are only O(1/n).

Feynman Path Integral Using Operator Integration in Banach Space

Feynman-Path Integral in Banach Space: In 1940, R.P. Feynman attempted to find a mathematical representation to express quantum dynamics of the general form for a double-slit experiment. His intuition on several slits with several walls in terms of Lagrangian instead of Hamiltonian resulted in a magnificent work. It was known as Feynman Path Integrals in quantum physics, and a large part of the scientific community still considers them a heuristic tool that lacks a sound mathematical definition. This paper aims to refute this prejudice, by providing an extensive and self-contained description of the mathematical theory of Feynman Path Integration, from the earlier attempts to the latest developments, as well as its applications to quantum mechanics. About a hundred years after the beginning of modern physics, it was realized that light could in fact show behavioral characteristics of both waves and particles. In 1927, Davisson and Germer demonstrated that electrons show the same dual behavior, which was later extended to atoms and molecules. We shall follow the method of integration with some modifications to construct a generalized Lebesgue-Bochner-Stieltjes (LBS) integral of the form , where u is a bilinear operator acting in the product of Banach spaces, f is a Bochner summable function, and μ is a vector-valued measure. We will demonstrate that the Feynman Path Integral is consistent and can be justified mathematically with LBS integration approach.

THE RELATION BETWEEN SIGN-CHANGING SOLUTION AND POSITIVE-NEGATIVE SOLUTIONS FOR NONLINEAR OPERATOR EQUATIONS AND ITS APPLICATIONS

By fixed point index theory and a result obtained by Amann, existence of the solution for a class of nonlinear operator equations x = Ax is discussed. Under suitable conditions, a couple of positive and negative solutions are obtained. Finally, the abstract result is applied to nonlinear Sturm-Liouville boundary value problem, and at least four distinct solutions are obtained.

Separability for Positive Operators on Tensor Product of Hilbert Spaces

The separability and the entanglement(that is,inseparability)of the composite quantum states play important roles in quantum information theory.Mathematically,a quantum state is a trace-class positive operator with trace one acting on a complex separable Hilbert space.In this paper,in more general frame,the notion of separability for quantum states is generalized to bounded positive operators acting on tensor product of Hilbert spaces.However,not like the quantum state case,there are different kinds of separability for positive operators with different operator topologies.Four types of such separability are discussed;several criteria such as the finite rank entanglement witness criterion,the positive elementary operator criterion and PPT criterion to detect the separability of the positive operators are established;some methods to construct separable positive operators by operator matrices are provided.These may also make us to understand the separability and entanglement of quantum states better,and may be applied to find new separable quantum states.

Young's Inequality for Positive Operators

The classical Young’s inequality and its refinements are applied to positive operators on a Hilbert space at first. Based on the classical Poisson integral formula of relevant operators, some new inequalities on unitarily invariant norm of A1-p XB1-q - A1-q Y B1-p are obtained with effective calculation, where A, B, X, Y ∈ B（H） with A, B 0 and 1 p, q ∞ with the conjugate exponent q = p/（p - 1）.

Application Of Jessen＇s Type Inequality For Positive C0-Semigroup Of Operators

Recently in [4], the Jessen＇s type inequality for normalized positive C0-semigroups is obtained. In this note, we present few results of this inequality, yielding Holder＇s Type and Minkowski＇s type inequalities for corresponding semigroup. Moreover, a Dresher＇s type inequality for two-parameter family of means, is also proved.

REFINEMENTS OF THE NORM OF TWO ORTHOGONAL PROJECTIONS

In this paper,some refinements of norm equalities and inequalities of combination of two orthogonal projections are established.We use certain norm inequalities for positive contraction operator to establish norm inequalities for combination of orthogonal projections on a Hilbert space.Furthermore,we give necessary and sufficient conditions under which the norm of the above combination of o`rthogonal projections attains its optimal value.

EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO A CLASS OF SEMILINEAR ELLIPTIC SYSTEMS

In this paper an existence and uniqueness theorem of positive solutions to a class of semilinear elliptic systems is proved. Also, a necessary condition for the existence of the positive solution is obtained. As the application of the main theorem, two examples are given.

ASYMPTOTIC APPROXIMATION OF FUNCTIONS AND THEIR DERIVATIVES BY GENERALIZED BASKAKOV-SZAZS-DURRMEYER OPERATORS

We present the complete asymptotic expansion for a generalization of the Baskakov-Szasz-Durrmeyer operators and their derivatives.

Association between operative position and postoperative nausea and vomiting in patients undergoing laparoscopic sleeve gastrectomy

BACKGROUND Bariatric surgery is one of the most effective ways to treat morbid obesity,and postoperative nausea and vomiting(PONV)is one of the common complications after bariatric surgery.At present,the mechanism of the high incidence of PONV after weight-loss surgery has not been clearly explained,and this study aims to investigate the effect of surgical position on PONV in patients undergoing bariatric surgery.AIM To explore the effect of the operative position during bariatric surgery on PONV.METHODS Data from obese patients,who underwent laparoscopic sleeve gastrectomy(LSG)in the authors’hospital between June 2020 and February 2022 were divided into 2 groups and retrospectively analyzed.Multivariable logistic regression analysis and the t-test were used to study the influence of operative position on PONV.RESULTS There were 15 cases of PONV in the supine split-leg group(incidence rate,50%)and 11 in the supine group(incidence rate,36.7%)(P=0.297).The mean operative duration in the supine split-leg group was 168.23±46.24 minutes and 140.60±32.256 minutes in the supine group(P<0.05).Multivariate analysis revealed that operative position was not an independent risk factor for PONV(odds ratio=1.192,95%confidence interval:0.376-3.778,P=0.766).CONCLUSION Operative position during LSG may affect PONV;however,the difference in the incidence of PONV was not statistically significant.Operative position should be carefully considered for obese patients before surgery.

AN ORDER PRESERVING INEQUALITY FOR THREE OPERATORS VIA FURUTA INEQUALITY

Furuta showed that if A≥B≥0,then for each r≥0,f（p）=（A^r/2 B^p A^r/2）^t＋r/p＋r is decreasing for p≥t≥0.Using this result,the following inequality（C^r/2（AB^2A）^δC^ r/2）^ p-1＋r/4δ＋r ≤C^p-1＋r is obtained for 0〈p ≤1,r≥1,1/4≤δ≤1 and three positive operators A, B, C satisfy（A^1/2BA^1/2）^p/2≤A^p,（B^1/2AB^1/2）^p/2≥B^p,（C^1/2AC^1/2）^p/2≤C^p,（A^1/2CA^1/2）^p/2≥A^p.

A Note on Generalized Aluthge Transformation for p-hyponormal and Log-hyponormal Operators

We shall give some results on generalized aluthge transformation for phyponormal and log-hyponormal operators. We shall also discuss the best possibility of these results.

Approximation of Generalized Szász-Mirakjan Operator

In this paper, the author define a kind of generalized Szasz-Mirakjan operator and discuss its convergence and degree of the approximation,extend some results got by J. Grof[1] and Z. Ditzian[2].

Generalized Szász-Mirakjan Operator Approximation

In this paper we shall defin a kind of generahzed Szász-Mirakjan operator and discuss its convergence and degree of approximation,extend some results got by J.Grof and Z.Ditzian.

Further Extension of the Relations between Two Operator Inequalities

In this paper,we give out a further extension of the relations between two famous inequalities(B r 2 A p B r 2) r p+r≥B r and Ap≥(A p 2 B r A p 2) p p+r,which can yield the result of Yamazaki and Yanagida's.

Approximation for Certain Stancu Type Summation Integral Operator

In the present paper, we consider Stancu type generalization of the summation integral type operators discussed in [15]. We apply hypergeometric series for obtaining moments of these operators. We also discuss about asymptotic formula and error estimation in terms of modules of continuity.

Evaluation index system for positive operation of water conservancy projects

The conditions for the positive operation of water conservancy projects are described in this paper. A scientific and effective evaluation index system was established based on frequency analysis, theoretical analysis, and expert consultation. This evaluation index system can be divided into six first-level indices： the degree to which facilities are intact and functionality standards are reached, the status of operation and management funds, the rationality and degree of advancement of the management team structure, the adaptability and rationality of the water conservancy project management system, the degree of automatization and informationization of the management techniques, and the conduciveness of the exterior environment. The weights for evaluation indices were obtained through the analytic hierarchy process method with consideration of the difference between public welfare and profit-oriented water conservancy projects. This study provides a scientific method for evaluating the positive operation of water conservancy projects.