Unitary invariants in multivariable operator theory

特征函数的 Sz.-Nagy 和 Foiaş 分解的正则性及其多变量类似物

Regularity of the Sz.-Nagy and Foiaş Factorization of Characteristic Functions and Its Multivariable Analogue

研究具有不变子空间的压缩算子和行压缩算子特征函数分解的正则性，构造反例说明一般非正则，给出正则性刻画，确定几类正则算子，引入\(k -\)奇异分解概念并证明特定压缩算子特征函数不存在\(k -\)奇异分解。

Isomorphism Invariants for Multivariable C*-Dynamics

To a given multivariable C*-dynamical system $(A, \al)$ consisting of *-automorphisms, we associate a family of operator algebras $\alg(A, \al)$, which includes as specific examples the tensor algebra and the semicrossed product. It is shown that if two such operator algebras $\alg(A, \al)$ and $\alg(B, \be)$ are isometrically isomorphic, then the induced dynamical systems $(\hat{A}, \hat{\al})$ and $(\hat{B}, \hat{\be})$ on the Fell spectra are piecewise conjugate, in the sense of Davidson and Katsoulis. In the course of proving the above theorem we obtain several results of independent inter…

On free resolutions in multivariable operator theory

We define the notion of a free resolution of a d-tuple $(T_1, T_2, . . . T_d)$ of mutually commuting operators acting on a Hilbert space H, and that this invariant gives rise to a class of vector space complexes parametrized by points in the unit ball $B_d = {z \in C^d: |z| <1}$. We show that for $λ\in B_d$ the homology of the corresponding complex is equivalent to the homology of the Koszul complex of a Mobius transform of $(T_1, T_2, . . . T_d)$. The notion of a Mobius transform in multivariable operator is defined, and some of its properties are investigated.

Curvature invariant and generalized canonical operator models - II

In [11] the authors investigated a family of quotient Hilbert modules in the Cowen-Douglas class over the unit disk constructed from classical Hilbert modules such as the Hardy and Bergman modules. In this paper we extend the results to the multivariable case of higher multiplicity. Moreover, similarity as well as isomorphism results are obtained.

Non-linear traces on matrix algebras, majorization, unitary invariant norms and 2-positivity

We study non-linear traces of Choquet type and Sugeno type on matrix algebras. They have certain partial additivities. We show that these partial additivities characterize non-linear traces of both Choquet type and Sugeno type respectively. There exists a close relation among non-linear traces of Choquet type, majorization, unitary invariant norms and 2-positivity.

The curvature invariant of a Hilbert module over C[z_1,...,z_d]

A notion of curvature is introduced in multivariable operator theory and an analogue of the Gauss-Bonnet-Chern theorem is established for graded (contractive) Hilbert modules over the complex polynomial algebra in d variables, d=1,2,3,.... The curvature invariant, Euler characteristic, and degree are computed for some explicit examples based on varieties in (multidimensional) complex projective space, and applications are given to the structure of graded ideals in C[z_1,...,z_d] and to the existence of "inner sequences" for closed submodules of the free Hilbert module H^2(C^d).