p-Adic valued quantization

This review covers an important domain of p-adic mathematical physics — quantum mechanics with p-adic valued wave functions. We start with basic mathematical constructions of this quantum model: Hilbert spaces over quadratic extensions of the field of p-adic numbers ? _p , operators — symmetric, unitary, isometric, one-parameter groups of unitary isometric operators, the p-adic version of Schrödinger’s quantization, representation of canonical commutation relations in Heisenberg andWeyl forms, spectral properties of the operator of p-adic coordinate.We also present postulates of p-adic valued quantization. Here observables as well as probabilities take values in ? _p . A physical interpretation of p-adic quantities is provided through approximation by rational numbers.     

Strongly irreducible operators on Banach spaces

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w*-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.     

A Regularization of Quantum Field Hamiltonians with the Aid of p–adic Numbers

Gaussian distributions on infinite-dimensional p-adic spaces are introduced and the corresponding L₂-spaces of p-adic-valued square integrable functions are constructed. Representations of the infinite-dimensional Weyl group are realized in p-adic L₂-spaces. There is a formal analogy with the usual Segal representation. But there is also a large topological difference: parameters of the p-adic infinite-dimensional Weyl group are defined only on some balls (these balls are additive subgroups). p-adic Hilbert space representations of quantum Hamiltonians for systems with an infinite number of degrees of freedom are constructed. Many Hamiltonians with potentials which are too singular to exist as functions over reals are realized as bounded symmetric operators in L₂-spaces with respect to a p-adic Gaussian distribution.     

p-adic 域上的拟微分算子

本文研究p-adic 域Q_p 上的一类拟微分算子{T^α : α∈R}, 其中算子T^α 在Q_p 的检验函数空 间S(Q_p) 以及相应的分布空间S′(Q_p) 中作为一个运算是封闭的. 本文构造了算子T^α 的卷积核, 指 出算子与其卷积核在解决p-adic 域上的相关问题中所起的重要作用. 最后研究算子T^α 的谱性质, 构 造了算子T^α 的全体固有值与固有函数, 并证明全体固有函数所成的集合组成L²(Q_p) 空间中的一组 完整正交基.     

Convolution factorability of bilinear maps and integral representations

In this paper we consider a special class of continuous bilinear operators acting in a product of Banach algebras of integrable functions with convolution product. In the literature, these bilinear operators are called ‘zero product preserving’, and they may be considered as a generalization of Lamperti operators. We prove a factorization theorem for this class, which establishes that each zero product preserving bilinear operator factors through a subalgebra of absolutely integrable functions. We obtain also compactness and summability properties for these operators under the assumption of some classical properties for the range spaces, as the Dunford–Pettis property or the Schur property and we give integral representations by some concavity properties of operators. Finally, we give some applications for integral transforms, and an integral representation for Hilbert–Schmidt operators.     

A note on the star order in Hilbert spaces

We study the star order on the algebra L(?) of bounded operators on a Hilbert space ?. We present a new interpretation of this order which allows to generalize to this setting many known results for matrices: functional calculus, semi-lattice properties, shorted operators and orthogonal decompositions. We also show several properties for general Hilbert spaces regarding the star order and its relationship with the functional calculus and the polar decomposition, which were unknown even in the finite-dimensional setting. We also study the existence of strong limits of star-monotone sequences and nets.     

Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups

A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group R ⁿ , the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrödinger operator based on Schwinger's approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621–648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrödinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used.     

Selfadjoint operators in S-spaces

We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by , where U is a unitary operator in . We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well-know and in general unsolved problem.     

On Fredholm Operators Between Non-archimedean Fréchet Spaces

It is proved that the index of a Fredholm operator between non-Archimedean Fréchet spaces is preserved under compact perturbations. A similar result is shown for Fredholm operators between non-Archimedean polar regular LF-spaces.     

Q_p上的共轭温度系

本文用于构造p-adic共轭温度系.首先,说明了热核及其Hilbert变换所适合的估计,描述了它们的正则性.并且对热核及其Hilbert变换在各个方向的导数进行了估计.然后,利用热核的卷积理论,得到了共轭温度系的边值特性.最后,通过共轭温度系解释了Hardy空间.     

空间H中G-框架的扰动

本文运用算子理论方法，讨论了Hilbert 空间$H$中$g$-框架和$g$-框架算子的性质; 并且研究了$g$-框架的扰动，给出了一些有意义的结果.     

Solutions of Tikhonov Functional Equations and Applications to Multiplication Operators on Szegö Spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert–Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.     

NON-ARCHIMEDEAN PROBABILITY: FREQUENCY AND AXIOMATICS THEORIES

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Approximators to generalized inverses as regularizers of ill-posed problems

Let H₁ and H₂ be Hilbert spaces and let B be a closed linear operator mapping a dense subset of H₁ into H₂. Several families of approximators which converge t o the orthogonal or Moore-Penrose generalized inverse B^? are constructed. Additionally, the approximators are shown t o provide regularization operators for the equation Bx=y. Some of the results are extended to dissipative operators on reflexive and general Banach spaces.     

Self‐adjoint Sturm–Liouville problems with an infinite number of boundary conditions

We study Sturm–Liouville (SL) problems on an infinite number of intervals, adjacent endpoints are linked by means of boundary conditions, and characterize the conditions which determine self‐adjoint operators in a Hilbert space which is the direct sum of the spaces for each interval. These conditions can be regular or singular, separated or coupled. Furthermore, the inner products of the summand spaces may be multiples of the usual inner products with different spaces having different multiples. We also extend the GKN Theorem to cover the infinite number of intervals theory with modified inner products and discuss the connection between our characterization and the classical one with the usual inner products. Our results include the finite number of intervals case.     

Stability of discrete-time positive evolution operators on ordered Banach spaces and applications

In this paper we discuss stability problems for a class of discrete-time evolution operators generated by linear positive operators acting on certain ordered Banach spaces. Our approach is based upon a new representation result that links a positive operator with the adjoint operator of its restriction to a Hilbert subspace formed by sequences of Hilbert–Schmidt operators. This class includes the evolution operators involved in stability and optimal control problems for linear discrete-time stochastic systems. The inclusion is strict because, following the results of Choi, we have proved that there are positive operators on spaces of linear, bounded and self-adjoint operators which have not the representation that characterize the completely positive operators. As applications, we introduce a new concept of weak-detectability for pairs of positive operators, which we use to derive sufficient conditions for the existence of global and stabilizing solutions for a class of generalized discrete-time Riccati equations. Finally, assuming weak-detectability conditions and using the method of Lyapunov equations we derive a new stability criterion for positive evolution operators.     

Partial inner product spaces. II. Operators

We study linear operators between nondegenerate partial inner product spaces and their relationships to selfadjoint operators in a “middle” Hilbert space.     

Difference operators,Green's matrix,sampling theory and applications in signal processing

This paper introduces sampling representations for discrete signals arising from self adjoint difference operators with mixed boundary conditions. The theory of linear operators on finite-dimensional inner product spaces is employed to study the second-order difference operators. We give necessary and sufficient conditions that make the operators self adjoint. The equivalence between the difference operator and a Hermitian Green's matrix is established. Sampling theorems are derived for discrete transforms associated with the difference operator. The results are exhibited via illustrative examples, involving sampling representations for the discrete Hartley transform. Families of discrete fractional Fourier-type transforms are introduced with an application to image encryption.     

Special bases for solutions of a generalized Maxwell system in 3‐dimensional space

The aim of this paper is to study complete polynomial systems in the kernel space of conformally invariant differential operators in higher spin theory. We investigate the kernel space of a generalized Maxwell operator in 3‐dimensional space. With the already known decomposition of its homogeneous kernel space into 2 subspaces, we investigate first projections from the homogeneous kernel space to each subspace. Then, we provide complete polynomial systems depending on the given inner product for each subspace in the decomposition. More specifically, the complete polynomial system for the homogenous kernel space is an orthogonal system wrt a given Fischer inner product. In the case of the standard inner product in L² on the unit ball, the provided complete polynomial system for the homogeneous kernel space is a partially orthogonal system. Further, if the degree of homogeneity for the respective subspaces in the decomposed kernel spaces approaches infinity, then the angle between the 2 subspaces approaches π/2.     

一类基于量子程序理论的序列效应代数

空间上的算子理论是量子力学的基本数学框架之一.Hilbert空间效应代数是指小于等于单位算子的正算子集合.我们引入了Hilbert空间效应代数的一类子序列效应代数，并讨论了其上序列积的基本运算性质.我们发现：由于代数结构的不同，这类新的序列效应代数与现有效应代数上的运算性质有很大差异.