线性算子和线性关系的一个标准分解

耗散线性关系的典范分解

Canonical decomposition of dissipative linear relations

本文利用 Sz. Nagy-Foiaş-Langer 与 von Neumann-Wold 分解，通过 Z 变换构造耗散线性关系的两种典范分解，实现自伴部分与完全非自伴部分的分离。

Canonical decomposition of operators associated with the symmetrized polydisc

A tuple of commuting operators $(S_1,\dots,S_{n-1},P)$ for which the closed symmetrized polydisc $Γ_n$ is a spectral set is called a $Γ_n$-contraction. We show that every $Γ_n$-contraction admits a decomposition into a $Γ_n$-unitary and a completely non-unitary $Γ_n$-contraction. This decomposition is an analogue to the canonical decomposition of a contraction into a unitary and a completely non-unitary contraction. We also find new characterizations for the set $Γ_n$ and $Γ_n$-contractions.

A canonical decomposition of strong $L^2$-functions

The aim of this paper is to establish a canonical decomposition of operator-valued strong $L^2$-functions by the aid of the Beurling-Lax-Halmos Theorem which characterizes the shift-invariant subspaces of vector-valued Hardy space. This decomposition invites us to coin a new notion of the "Beurling degree" of the inner function. Eventually, we establish a deep connection between the spectral multiplicity of the model operator and the Beurling degree of the corresponding characteristic function.

Canonical decomposition of a tetrablock contraction and operator model

A triple of commuting operators for which the closed tetrablock $\overline{\mathbb E}$ is a spectral set is called a tetrablock contraction or an $\mathbb E$-contraction. The set $\mathbb E$ is defined as \[ \mathbb E = \{ (x_1,x_2,x_3)\in\mathbb C^3\,:\, 1-zx_1-wx_2+zwx_3\neq 0 \textup{ whenever } |z|\leq 1, |w|\leq 1 \}. \] We show that every $\mathbb E$-contraction can be uniquely written as a direct sum of an $\mathbb E$-unitary and a completely non-unitary $\mathbb E$-contraction. It is analogous to the canonical decomposition of a contraction operator into a unitary and a completely non-…

Canonical Agler Decompositions and Transfer Function Realizations

A seminal result of Agler proves that the natural de Branges-Rovnyak kernel function associated to a bounded analytic function on the bidisk can be decomposed into two shift-invariant pieces. Agler's decomposition is non-constructive-a problem remedied by work of Ball-Sadosky-Vinnikov, which uses scattering systems to produce Agler decompositions through concrete Hilbert space geometry. This method, while constructive, so far has not revealed the rich structure shown to be present for special classes of functions-inner and rational inner functions. In this paper, we show that most of the impor…

On the Jordan decomposition for a class of non-symmetric Ornstein-Uhlenbeck operators

In this paper, we calculate the Jordan decomposition (or say, the Jordan canonical form) for a class of non-symmetric Ornstein-Uhlenbeck operators with the drift coefficient matrix being a Jordan block and the diffusion coefficient matrix being identity multiplying a constant. For the 2-dimensional case, we present all the general eigenfunctions by the induction. For the 3-dimensional case, we divide the calculating of the Jordan decomposition into several steps (the key step is to do the canonical projection onto the homogeneous Hermite polynomials, next we use the theory of systems of linear…