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University of Macau / Faculty of Science and Technology / Department of Mathematics

University of Macau / Faculty of Science and Technology / Department of Mathematics

A pair of commuting bounded operators (S, P) for which the set r = {(z 1 +z 2,z 1z 2) : |z 1| ≤1, |z 2| ≤1} C2 is a spectral set, is called a r-contraction in the literature. For a contraction P and a bounded commutant S of P, we seek a solution of the operator equation S –S*P = (I –P*P)½ X(I –P*P)½ where X is a bounded operator on Ran(I – P*P)½ with numerical radius of X being not greater than 1. We show the existence and uniqueness of solution to the operator equation above when (S,P) is a r-contraction. We call the unique solution, the fundamental operator of the r-contraction (S,P). As the title indicates, there are three parts of this thesis and the main role in all three parts is played by the fundamental operator. The existence of the fundamental operator allows us to explicitly construct a r-isometric dilation of a r-contraction (S,P), whereas its uniqueness guarantees the uniqueness of the minimal r-isometric dilation. The fundamental operator helps us to produce a genuine functional model for pure r-contractions. Also it leads us to a complete unitary invariant for pure r-contractions. We decipher the structures of r-isometries and r-unitaries by characterizing them in several different ways. We establish the fact that for every pure r-contraction (S,P), there is a bounded operator C with numerical radius being not greater than 1 such that S = C + C* P. When (S,P) is a r-isometry, S has the same form where P is an isometry commuting with C and C*. Also when (S,P) is a r-unitary, S has the same form too with P and C being commuting unitaries. Examples of r-contractions on reproducing kernel Hilbert spaces and their dilations are discussed.

Specific non-relativistic quantum mechanical one-dimensional systems, interacting via point interactions, are discussed within the resolvent algebra setting.

付記する学位プログラム名: 京都大学卓越大学院プログラム「先端光・電子デバイス創成学」 / 京都大学 / 新制・課程博士 / 博士(工学) / 甲第25290号 / 工博第5249号 / 新制||工||1999(附属図書館) / 京都大学大学院工学研究科電気工学専攻 / (主査)教授 阪本 卓也, 教授 引原 隆士, 准教授 薄 良彦, 教授 土居 伸二 / 学位規則第4条第1項該当 / Doctor of Agricultural Science / Kyoto University / DFAM

[pt] Um operador definido em um espaço de Hilbert é uniformemente estável se ele converge na topologia da norma para o operador nulo. O problema de Estabilidade Multiplicativa investiga quais são as classes de operadores que estabilizam uniformemente o operador original por uma perturbação multiplicativa. Neste trabalho colocamos este problema no contexto de produto tensorial e investigamos quais as classes que estabilizam multiplicativamente Contrações Fortemente Estáveis sob uma perturbação compacta. Em particular, apresentamos uma solução para o Problema de Estabilidade Multiplicativa para Contrações Fortemente Estáveis. / [en] An operator on a Hilbert space is uniformly stable if it converges to the null operator on the norm topology. The Multiplicative Stabilization Problem investigates which operators classes uniformly stabilize de original operator under multiplicative perturbation. This work consider the previous problem under the tensor product framework and investigates which operators classes multiplicative stabilize Strongly Stable Contraction under compact perturbations. We have established a solution to the Multiplicative Stabilization Problem for Strongly Stable Contractions.

The first part of the thesis studies invariant subspaces of strictly singular operators. By a celebrated result of Aronszajn and Smith, every compact operator has an invariant subspace. There are two classes of operators which are close to compact operators: strictly singular and finitely strictly singular operators. Pelczynski asked whether every strictly singular operator has an invariant subspace. This question was answered by Read in the negative. We answer the same question for finitely strictly singular operators, also in the negative. We also study Schreier singular operators. We show that this subclass of strictly singular operators is closed under multiplication by bounded operators. In addition, we find some sufficient conditions for a product of Schreier singular operators to be compact. The second part studies almost invariant subspaces. A subspace Y of a Banach space is almost invariant under an operator T if TY is a subspace of Y+F for some finite-dimensional subspace F ("error"). Almost invariant subspaces of weighted shift operators are investigated. We also study almost invariant subspaces of algebras of operators. We establish that if an algebra is norm closed then the dimensions of "errors" for the operators in the algebra are uniformly bounded. We obtain that under certain conditions, if an algebra of operators has an almost invariant subspace then it also has an invariant subspace. Also, we study the question of whether an algebra and its closure have the same almost invariant subspaces. The last two parts study collections of positive operators (including positive matrices) and their invariant subspaces. A version of Lomonosov theorem about dual algebras is obtained for collections of positive operators. Properties of indecomposable (i.e., having no common invariant order ideals) semigroups of nonnegative matrices are studied. It is shown that the "smallness" (in various senses) of some entries of matrices in an indecomposable semigroup of positive matrices implies the "smallness" of the entire semigroup. / Mathematics

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Let H be a separable Hilbert space over the complex field. The class S := {N|M : N is normal on H and M is an invariant subspace for Ng of subnormal operators. This notion was introduced by Halmos. The minimal normal extension Ň of a subnormal operator S was introduced by σ (S) and then Bram proved that Halmos. Halmos proved that σ(Ň) (S) is obtained by filling certain number of holes in the spectrum (Ň) of the minimal normal extension Ň of a subnormal operator S. Let σ (S) := σ (Ň) be the spectrum of the minimal normal extension Ň of S; which is called the normal spectrum of a subnormal operator S: This notion is due to Abrahamse and Douglas. We give several well-known characterization of subnormality. Let C* (S1) and C* (S2) be the C*- algebras generated by S1 and S2 respectively, where S1 and S2 are bounded operators on H: Next we give a characterization for subnormality which is purely C - algebraic. We also establish an intrinsic characterization of the normal spectrum for a subnormal operator, which enables us to answer the fol-lowing two questions. Let II be a *- representation from C* (S1) onto C* (S2) such that II(S1) = S2. If S1 is subnormal, then does it follow that S2 is subnormal? What is the relation between σ (S1) and σ (S2)? The first question was asked by Bram and second was asked by Abrahamse and Douglas. Answers to these questions were given by Bunce and Deddens.