一类无穷维Hamilton算子特征函数系的完备性

本文对分离变量后可转化为Sturm-Liouville问题的偏微分方程,引入Hamilton体系,从而导出无穷维Hamilton算子的特征值问题.然后利用辛空间的知识讨论了无穷维Hamilton算子特征函数系的完备性,为对此类方程应用基于Hamilton体系的分离变量法提供了理论基础.作为应用,还给出了波动方程导出的无穷维Hamilton算子特征函数系的完备性.  

Additive Results of the Drazin Inverse for Bounded Linear Operators

Given two bounded linear operators $P$ and $Q$ on a Banach space the formula for the Drazin inverse of $P+Q$ is given, under the assumptions $P^2 Q+PQ^2=0$ and $P^3 Q=PQ^3=0$ . In particular, some recent results arising in Drazin (Am Math Mon 65:506–514, 1958), Hartwig et al. (Linear Algebra Appl 322:207–217, 2001) and Castro-González et al. (J Math Anal Appl 350:207–215, 2009) are extended.  

Invertibility of Operator Matrices on a Banach Space

Let $\mathcal X $ and $\mathcal Y $ be Banach spaces, and let $A\in \mathcal B (\mathcal X )$ and $C\in \mathcal B (\mathcal Y , \mathcal X )$ be given operators. A necessary and sufficient condition is given for $\left[ \begin{array}{cc} A&C \\ X&Y \\ \end{array} \right]$ to be invertible (respectively, left invertible) for some $X\in \mathcal B (\mathcal X , \mathcal Y )$ and $Y\in \mathcal B (\mathcal Y )$ . Furthermore, some related results are obtained.  

On invertible nonnegative Hamiltonian operator matrices

Some new characterizations of nonnegative Hamiltonian operator matrices are given. Several necessary and sufficient conditions for an unbounded nonnegative Hamiltonian operator to be invertible are obtained, so that the main results in the previously published papers are corollaries of the new theorems. Most of all we want to stress the method of proof. It is based on the connections between Pauli operator matrices and nonnegative Hamiltonian matrices.  

Invertibility of nonnegative Hamiltonian operator with unbounded entries

In this paper, the invertibility of nonnegative Hamiltonian operator with unbounded entries is studied, and the sufficient conditions for the everywhere defined bounded invertibility of nonnegative Hamiltonian operator are obtained.  

Spectral inclusion properties of the numerical range in a space with an indefinite metric

In this paper the numerical range of operators (possibly unbounded) in an indefinite inner product space is studied. In particular, we show that the spectrums of bounded positive operators (or the spectrum of unbounded uniformly I-positive operators) are contained in the closure of the I-numerical range.  

On the Right (Left) Invertible Completions for Operator Matrices

Let ${\mathcal {H}_{1}}Let H₁{\mathcal {H}_{1}} and H₂{\mathcal {H}_{2}} be separable Hilbert spaces, and let A ? B(H₁), B ? B(H₂){A \in \mathcal {B}(\mathcal {H}_{1}),\, B \in \mathcal {B}(\mathcal {H}_{2})} and C ? B(H₂, H₁){C \in \mathcal {B}(\mathcal {H}_{2},\, \mathcal {H}_{1})} be given operators. A necessary and sufficient condition is given for ${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)}${\left(\begin{smallmatrix}A &\enspace C\\ X &\enspace B \end{smallmatrix}\right)} to be a right (left) invertible operator for some X ? B(H₁, H₂){X \in \mathcal {B}(\mathcal {H}_{1},\, \mathcal {H}_{2})}. Furthermore, some related results are obtained.  

Invertibility of 2 × 2 operator matrices

Properties of right invertible row operators, i.e., of 1 × 2 surjective operator matrices are studied. This investigation is based on a specific space decomposition. Using this decomposition, we characterize the invertibility of a 2 × 2 operator matrix. As an application, the invertibility of Hamiltonian operator matrices is investigated.  

Symplectic Self-adjointness of Infinite Dimensional Hamiltonian Operators

Symplectic self-adjointness of infinite dimensional Hamiltonian operators is studied, the necessary and sufficient conditions are given. Using the relatively bounded perturbation, the sufficient conditions about symplectic self-adjointness are shown.