Literatura académica sobre el tema "Symmetrized bidisk"

碩士<br>東海大學<br>數學系<br>89<br>Consider symmetrized bidisc $\Gamma_{2}$:% $$\Gamma_{2}\triangleq \{(s,p):\lambda^{2}-s\lambda+p=0,~\lambda \in \mathbb{C},~|\lambda|\leq1\}$$% and spectral Nevanlinna-Pick Interpolation non-flat problem on it as:\\ % Given $\alpha_{1},~\alpha_{2} \in \mathbb{D},~(s_{1},0),~(s_{2},0) % \in {\rm Int}~\Gamma_{2} $,% $\varphi : \mathbb{D} \longrightarrow {\rm Int}~\Gamma_{2}$,is analytic,% ~such that~$\varphi(\alpha_{1}) = (s_{1},0)$，$\varphi(\alpha_{2}) = (s_{2},~0)$,% ~by the equality of Carath$\acute{e}$odory and Kobayashi distances,% ~and Schur theorem, ~we can find $\varphi$ that we want.

碩士<br>東海大學<br>數學系<br>91<br>In this paper we discuss the two-point spectral Nevanlinna-Pick interpolation problem for 2 2 general case by using the previous results of T.D.Lin[13], C.T.Lin[8] and Yeh[9]: Given distinct ， ， ， ,find an analytic function such that and it's realization.

A pair of commuting bounded operators (S; P ) acting on a Hilbert space, is called a -contraction, if it has the symmetrised bides = f(z1 + z2; z1z2) : jz1j 1; jz2j 1g C2 as a spectral set. For every -contraction (S; P ), the operator equation S S P = DP F DP has a unique solution F 2 B(DP ) with numerical radius, denoted by w(F ), no greater than one, where DP is the positive square root of (I P P ) and DP = RanDP . This unique operator is called the fundamental operator of (S; P ). This thesis constructs an explicit normal boundary dilation for -contractions. A triple of commuting bounded operators (A; B; P ) acting on a Hilbert space with the tetra block E = f(a11; a22; detA) : A = a11 a12 with kAk 1g C 3 a21 a22 as a spectral set, is called a tetra block contraction. Every tetra block contraction possesses two fundamental operators and these are the unique solutions of A B P = DP F1DP ; and B A P = DP F2DP : Moreover, w(F1) and w(F2) are no greater than one. This thesis also constructs an explicit normal boundary dilation for tetra block contractions. In these constructions, the fundamental operators play a pivotal role. Both the dilations in the symmetrised bidisc and in the tetra block are proved to be minimal. But unlike the one variable case, uniqueness of minimal dilations fails in general in several variables, e.g., Ando's dilation is not unique, see [44]. However, we show that the dilations are unique under a certain natural condition. In view of the abundance of operators and their complicated structure, a basic problem in operator theory is to find nice functional models and complete sets of unitary invariants. We develop a functional model theory for a special class of triples of commuting bounded operators associated with the tetra block. We also find a set of complete unitary invariants for this special class. Along the way, we find a Burling-Lax-Halmos type of result for a triple of multiplication operators acting on vector-valued Hardy spaces. In both the model theory and unitary invariance, fundamental operators play a fundamental role. This thesis answers the question when two operators F and G with w(F ) and w(G) no greater than one, are admissible as fundamental operators, in other words, when there exists a -contraction (S; P ) such that F is the fundamental operator of (S; P ) and G is the fundamental operator of (S ; P ). This thesis also answers a similar question in the tetra block setting.

碩士<br>東海大學<br>數學系<br>98<br>The symmetrrized bidisc is defined as the set of two coefficients of a quadratic equation with its roots located inside the unit disc. In this thesis, a matlab-based GUI is developed to the graphs of the symmetrized bidisc and associated spectral interpolating functions. Since the symmetrized bidisc belongs to C^2, its 3D projection is plotted as the real or imaginary part of one variable is fixed. By the way, the graph of the symmetrized bidisc is also shown when the radius of the root's location changes. Furthermorre, two kinds of approaches are used to construct the spectral interoplating function defined on the symmetrized bidisc are introduced and their graphs are depicted as well. Once the interpolating function is computed, we demo how to construct the interpolation function to solve the two-by-two spectral Nevanlinna-Pick problem. Keywords: unit disc, symmetrized bidisc, quadratic equation, matlab, GUI, spectral Nevanlinna- Pick interpolation problemn