套子代数上线性映射的Lie不变子空间

本文引入了Banach代数上线性映射的Lie不变子空间,给出了因子VonNeumann代数中套子代数上以导子空间为Lie不变子空间的线性映射的一般形式,研究了Lie导子与Lie自同构的概念及了Lie导子与Lie自同构半群的关系.     

CPT-Frames for Non-Hermitian Hamiltonians

Based on the P T-symmetric quantum theory,the concepts of P T-frame,P T-symmetric operator and CPT-frame on a Hilbert space K and for an operator on K are proposed.It is proved that the spectrum and point spectrum of a P T-symmetric linear operator are both symmetric with respect to the real axis and the eigenvalues of an unbroken P T-symmetric operator are real.For a linear operator H on Cd,it is proved that H has unbroken P Tsymmetry if and only if it has d diferent eigenvalues and the corresponding eigenstates are eigenstates of P T.Given a C P T-frame on K,a new positive inner product on K is induced and called C P T-inner product.Te relationship between the CP T-adjoint and the Dirac adjoint of a densely defined linear operator is derived,and it is proved that an operator which has a bounded CP T-frame is CP T-Hermitian if and only if it is T-symmetric,in that case,it is similar to a Hermitian operator.The existence of an operator C consisting of a CP T-frame is discussed.These concepts and results will serve a mathematical discussion about P T-symmetric quantum mechanics.     

Hilbert空间中有效序列的刻画与扰动

本文研究了Hilbert空间中的有效序列,通过引入序列的伴随序列,给出了有效序列的一个等价刻画,接着讨论了保持向量序列有效性的线性算子的性质,证明了一个线性算子将任意有效序列映射为有效序列当且仅当它是酉算子;最后,给出了正规正交基中添加一个单位向量后所得到的序列是有效序列的一个充要条件.     

效应代数的表示

本文讨论了抽象效应代数的表示问题. 对于一个抽象效应代数(E,⊕, 0, 1), 如果存在一个Hilbert 空间 H 和一个单态射 φ:E →ε(H), 那么称 E 为可表示的且称(φ,H) 是E 的一个表示, 其中ε(H) 表示 H 上所有正压缩算子构成的效应代数. 给出了一些可表示的和不可表示的效应代数的例子, 证 明了非空集 X 上的任一模糊集系统 F 和Boolean 代数B_X 都是可表示的效应代数.     

Hilbert空间中的Bessel列构成的 $C^*$-代数 $B_H(I)$

Let H be a separable Hilbert space, B H(I), B(H) and K(H) the sets of all Bessel sequences {f i}i∈I in H, bounded linear operators on H and compact operators on H, respectively. Two kinds of multiplications and involutions are introduced in light of two isometric linear isomorphisms αH : B H(I) → B(?2), β : B H(I) → B(H), respectively, so that B H(I) becomes a unital C*-algebra under each kind of multiplication and involution. It is proved that the two C*-algebras(B H(I), ?, ?) and(B H(I), ·, *) are *-isomorphic. It is also proved that the set F H(I) of all frames for H is a unital multiplicative semi-group and the set R H(I) of all Riesz bases for H is a self-adjoint multiplicative group, as well as the set K H(I) := β-1(K(H)) is the unique proper closed self-adjoint ideal of the C*-algebra B H(I).     

Digraph states and their neural network representations

With the rapid development of machine learning, artificial neural networks provide a powerful tool to represent or approximate many-body quantum states. It was proved that every graph state can be generated by a neural network. Here, we introduce digraph states and explore their neural network representations (NNRs). Based on some discussions about digraph states and neural network quantum states (NNQSs), we construct explicitly an NNR for any digraph state, implying every digraph state is an NNQS. The obtained results will provide a theoretical foundation for solving the quantum many-body problem with machine learning method whenever the wave-function is known as an unknown digraph state or it can be approximated by digraph states.