二次算子族与无穷维Hamilton算子的谱分析
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摘要
1990年,钟万勰院士将弹性力学导入Hamilton体系,建立了弹性力学求解新体系,将弹性力学与无穷维Hamilton算子相结合,提出了基于Hamilton体系的分离变量法,解决了许多实际问题.而基于Hamilton体系的分离变量法的理论基础便是无穷维Hamilton算子的谱理论及特征函数系的完备性.因此本学位论文以无穷维Hamilton算子的谱理论为中心,主要进行了如下几个方面的研究工作：第一.引入了一类有深刻力学背景的二次算子族L(λ)=λ2M—iλK—A,讨论了该类算子族的谱分布；第二.将无穷维Hamilton算子与二次算子族联系起来得到了无穷维Hamilton算子的谱分布；第三.讨论了二次算子族L(λ)=λ2M—iAK—A的数值域,证明了该类二次算子族的数值域关于虚轴对称；第四.研究了一类无穷维Hamilton算子的谱,利用无穷维Hamilton算子是算子矩阵且有特殊的结构给出了无穷维Hamilton算子谱的一种描述；最后,讨论了无穷维Hamilton算子的本质谱,利用Fredholm算子的定义,给出了无穷维Hamilton算子Fredholm意义下的本质谱的描述.
首先,我们讨论了一类在弦和梁的微小振动中出现的一类特殊的二次算子族L(λ)=λ2M—iλK—A的谱分布,给出了当M,K,A是非负算子时二次算子族L(λ)的谱位于上半开平面,上半闭平面的充分条件,同时给出了它位于下半平面上的谱都是纯虚谱的充分条件.
其次,利用二次算子族的谱分布讨论了无穷维Hamilton算子的谱分布.我们知道无穷维Hamilton算子的一个重要研究方向就是无穷维Hamilton算子生成算子半群的问题,要解决无穷维Hamilton算子生成半群的的问题就得首先解决无穷维Hamilton算子的谱分布.本文我们给出了非负无穷维Hamilton算子的谱位于上半开平面,上半闭平面的充分条件,同时给出了它位于下半平面上的谱都是纯虚谱的充分条件.
再次,我们知道算子的数值域是刻画算子谱的一种很有效的手段,因此我们讨论了二次算子族L(λ)=λ2M—iλK—A的数值域,证明了该类算子族的点谱是其数值域的一个子集,近似点谱是其数值域闭包的子集,同时我们证明了该类二次算子族的数值域是关于虚轴对称的.而其它类型的二次算子族却不具有这样的性质,本文也给出了例子加以说明.
接着,我们利用无穷维Hamilton算子的特殊结构,给出了无穷维Hamilton算子谱的描述.由于无穷维Hamilton算子是算子矩阵,我们利用其分块算子的组合形式给出了无穷维Hamilton算子谱的一种描述.
最后,我们讨论了无穷维Hamilton算子的本质谱,利用上半-Fredholm算子,下半-Fredholm算子及Fredholm算子的定义给出了无穷维Hamilton算子一类本质谱和三类本质谱的描述,利用其分块算子矩阵的特征,将其本质谱用其分块算子矩阵的组合形式来刻画.
本文研究了二次算子族及无穷维Hamilton算子的谱问题,为进一步研究无穷维Hamilton算子的相关问题做了一些理论上的铺垫,为进一步研究无穷维Hamilton系统做了一些有益的工作.
1990,Academician Wanxie Zhong conducted the elasticity to the infinite dimensional Hamiltonian system, established the new systematic methodology for the theory of elasticity.This method combined the elasticity with infinite di-mensional Hamiltonian operators,proposed the method of separation of variables based on Hamiltonian systems.It's valuable to many questions.lt is wellknown of us,the method of separation of variables on account of Hamiltonian systems was based on the spectral theory of the infinite dimensional Hamiltonian operators and its completeness of the eigenfunction systems.In this dissertation,we centere on the spectral theory of the infinite dimensional Hamiltonian operators,discuss some questions as follows:Firstly,we conducted a class of quadratic operator pen-cils L(λ)=λ2M-iλK-Aand discuss the scatter of its spectrum.Secondly,we re-search the scatter of the spectrum of the non-negative infinite dimensional Hamil-tonian operators.Thirdly,the numerical range of the quadratic operator pencils is discussed. Fourthly,we discuss the spectrum of a class of the infinite dimensional Hamiltonian operators.The last,we discuss the essential spectrum of a class of infinite dimensional Hamiltonian operators.
Firstly,we study the quadratic operator pencils L(λ)=λ2M-iλK-arising in researching the small vibrations of strings and beams.We give some sufficient conditions of the spectrum of this operator pencils.The sufficient conditions are when its spectrum located in the open upper half-plane and when its spectrum located in closed upper half-plane.And we give the condition of the spectrum which located in the open lower half-plane and the closed lower half-plane are only pure imaginary spectrum.
Secondly,we study the scatter of the infinite dimensional Hamiltonian opera-tor.Wellknown of us, it is important a infinite dimensional Hamiltonian operator to be a generator of an operator semigroup.But if we try to solve this problem,we must solve the scatter of the spectrum of the operators which we discussed.,So we study the distributions of a class of non-negative infinite dimensional Hamil-tonian operator in chapter3.We give the sufficient conditions when its spectrum located in the open upper half-plane and when its spectrum located in the closed upper half-plane.Meanwhile we give the conditions of the infinite dimensional Hamiltonian operator whose spectrum is pure imaginary spectrum which located in the open lower half-plane and the closed lower half-plane.
Thirdly,we discussed the numerical range of the quadratic operator pencils L(λ)=λ2M-iλK-A.Numerical range is a very useful method to estimate the spectrum of operator.To the numerical range of the quadratic operator pencils,we prove that its point spectrum is subset of its numerical range and its approximate spectrum is a subset of the closure of its numerical range.We prove its numerical is symmetric to the imaginary axis,but its not true to other operator pencils.We plot figures of numerical range of this quadratic operator pencils with Matlab software to discribe this result.
Fourthly,by the special structure of the infinite dimensional Hamiltonian op-erator,we give a necessary and sufficient condition of its spectrum description.we know that the infinite dimensional Hamiltonian operator is an operator matrix,so we describe its spectrum with its entries operator.We obtain that its spectrum is equal to some combination of its entries.
Lastly,we studied the essential spectrum of the infinite dimensional Hamilto-nian operator.We obtained the sufficient and necessary conditions of the infinite dimensional Hamiltonian operator to describe its spectrum.We give a description of the three classes of the essential spectrum of the infinite dimensional Hamil-tonian operator.
All in all,We mainly study the spectrum of the quadratic operator pencil and the infinite dimensional Hamiltonian operator.This research provide some theoretical foundations for the application of the infinite dimensional Hamiltonian operator.
首先,我们讨论了一类在弦和梁的微小振动中出现的一类特殊的二次算子族L(λ)=λ2M—iλK—A的谱分布,给出了当M,K,A是非负算子时二次算子族L(λ)的谱位于上半开平面,上半闭平面的充分条件,同时给出了它位于下半平面上的谱都是纯虚谱的充分条件.
其次,利用二次算子族的谱分布讨论了无穷维Hamilton算子的谱分布.我们知道无穷维Hamilton算子的一个重要研究方向就是无穷维Hamilton算子生成算子半群的问题,要解决无穷维Hamilton算子生成半群的的问题就得首先解决无穷维Hamilton算子的谱分布.本文我们给出了非负无穷维Hamilton算子的谱位于上半开平面,上半闭平面的充分条件,同时给出了它位于下半平面上的谱都是纯虚谱的充分条件.
再次,我们知道算子的数值域是刻画算子谱的一种很有效的手段,因此我们讨论了二次算子族L(λ)=λ2M—iλK—A的数值域,证明了该类算子族的点谱是其数值域的一个子集,近似点谱是其数值域闭包的子集,同时我们证明了该类二次算子族的数值域是关于虚轴对称的.而其它类型的二次算子族却不具有这样的性质,本文也给出了例子加以说明.
接着,我们利用无穷维Hamilton算子的特殊结构,给出了无穷维Hamilton算子谱的描述.由于无穷维Hamilton算子是算子矩阵,我们利用其分块算子的组合形式给出了无穷维Hamilton算子谱的一种描述.
最后,我们讨论了无穷维Hamilton算子的本质谱,利用上半-Fredholm算子,下半-Fredholm算子及Fredholm算子的定义给出了无穷维Hamilton算子一类本质谱和三类本质谱的描述,利用其分块算子矩阵的特征,将其本质谱用其分块算子矩阵的组合形式来刻画.
本文研究了二次算子族及无穷维Hamilton算子的谱问题,为进一步研究无穷维Hamilton算子的相关问题做了一些理论上的铺垫,为进一步研究无穷维Hamilton系统做了一些有益的工作.
1990,Academician Wanxie Zhong conducted the elasticity to the infinite dimensional Hamiltonian system, established the new systematic methodology for the theory of elasticity.This method combined the elasticity with infinite di-mensional Hamiltonian operators,proposed the method of separation of variables based on Hamiltonian systems.It's valuable to many questions.lt is wellknown of us,the method of separation of variables on account of Hamiltonian systems was based on the spectral theory of the infinite dimensional Hamiltonian operators and its completeness of the eigenfunction systems.In this dissertation,we centere on the spectral theory of the infinite dimensional Hamiltonian operators,discuss some questions as follows:Firstly,we conducted a class of quadratic operator pen-cils L(λ)=λ2M-iλK-Aand discuss the scatter of its spectrum.Secondly,we re-search the scatter of the spectrum of the non-negative infinite dimensional Hamil-tonian operators.Thirdly,the numerical range of the quadratic operator pencils is discussed. Fourthly,we discuss the spectrum of a class of the infinite dimensional Hamiltonian operators.The last,we discuss the essential spectrum of a class of infinite dimensional Hamiltonian operators.
Firstly,we study the quadratic operator pencils L(λ)=λ2M-iλK-arising in researching the small vibrations of strings and beams.We give some sufficient conditions of the spectrum of this operator pencils.The sufficient conditions are when its spectrum located in the open upper half-plane and when its spectrum located in closed upper half-plane.And we give the condition of the spectrum which located in the open lower half-plane and the closed lower half-plane are only pure imaginary spectrum.
Secondly,we study the scatter of the infinite dimensional Hamiltonian opera-tor.Wellknown of us, it is important a infinite dimensional Hamiltonian operator to be a generator of an operator semigroup.But if we try to solve this problem,we must solve the scatter of the spectrum of the operators which we discussed.,So we study the distributions of a class of non-negative infinite dimensional Hamil-tonian operator in chapter3.We give the sufficient conditions when its spectrum located in the open upper half-plane and when its spectrum located in the closed upper half-plane.Meanwhile we give the conditions of the infinite dimensional Hamiltonian operator whose spectrum is pure imaginary spectrum which located in the open lower half-plane and the closed lower half-plane.
Thirdly,we discussed the numerical range of the quadratic operator pencils L(λ)=λ2M-iλK-A.Numerical range is a very useful method to estimate the spectrum of operator.To the numerical range of the quadratic operator pencils,we prove that its point spectrum is subset of its numerical range and its approximate spectrum is a subset of the closure of its numerical range.We prove its numerical is symmetric to the imaginary axis,but its not true to other operator pencils.We plot figures of numerical range of this quadratic operator pencils with Matlab software to discribe this result.
Fourthly,by the special structure of the infinite dimensional Hamiltonian op-erator,we give a necessary and sufficient condition of its spectrum description.we know that the infinite dimensional Hamiltonian operator is an operator matrix,so we describe its spectrum with its entries operator.We obtain that its spectrum is equal to some combination of its entries.
Lastly,we studied the essential spectrum of the infinite dimensional Hamilto-nian operator.We obtained the sufficient and necessary conditions of the infinite dimensional Hamiltonian operator to describe its spectrum.We give a description of the three classes of the essential spectrum of the infinite dimensional Hamil-tonian operator.
All in all,We mainly study the spectrum of the quadratic operator pencil and the infinite dimensional Hamiltonian operator.This research provide some theoretical foundations for the application of the infinite dimensional Hamiltonian operator.
引文
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