Coupling and relaxed commutant lifting

非交换交换子提升、插值与移位不变子空间

Noncommutative commutant lifting, interpolation, and shift-invariant subspaces

本文在非交换Drury-Arveson空间中建立Sarason型交换子提升定理，解决非交换Nevanlinna-Pick插值问题，并给出移位不变子空间的Beurling-Lax-Halmos型刻画。

等距膨胀与多变量中的Sarason交换子提升定理

Isometric dilation and Sarason's commutant lifting theorem in several variables

本文针对n≥3的可交换压缩算子元组，构造BCL型等距膨胀，证明von Neumann不等式及Sarason型交换子提升定理，并发现新的可膨胀算子类。

Relaxed commutant lifting: an equivalent version and a new application

This paper presents a few additions to commutant lifting theory. An operator interpolation problem is introduced and shown to be equivalent to the relaxed commutant lifting problem. Using this connection a description of all solutions of the former problem is given. Also a new application, involving bounded operators induced by $H^2$ operator-valued functions, is presented.

All solutions to the relaxed commutant lifting problem

A new description is given of all solutions to the relaxed commutant lifting problem. The method of proof is also different from earlier ones, and uses only an operator-valued version of a classical lemma on harmonic majorants.

Agler-Commutant Lifting on an Annulus

The main result is a test function style commutant lifting theorem for an annulus A. The test functions are the minimal inner functions for A. The model space is the Sarason Hardy Hilbert space for A uniquely determined by the fact that its reproducing kernel has no zeros.

Bi-isometries and commutant lifting

A new functional model for pairs of commuting isometries is described. Intertwining operators between such models are then studied in order to approach the classification of invariant subspaces of such pairs.

The inverse commutant lifting problem: characterization of associated Redheffer linear-fractional maps

It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identifies solutions of the lifting problem with minimal unitary extensions of a partially defined isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefficient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitar…

多圆盘上的换位提升、插值与扰动

Commutant lifting, interpolation, and perturbations on the polydisc

针对Sarason换位提升基本定理不适用于多圆盘的问题，本文提出多变量换位提升的两种分类，解决了多圆盘上有界解析函数的Nevanlinna-Pick插值与扰动问题，推进了希尔伯特函数空间理论的两个著名问题。

Commutant Lifting for Commuting Row Contractions

If $T= \big[ T_1 ... T_n\big]$ is a row contraction with commuting entries, and the Arveson dilation is $\tilde T= \big[ \tilde T_1 ... \tilde T_n\big]$, then any operator $X$ commuting with each $T_i$ dilates to an operator $Z$ of the same norm which commutes with each $\tilde T_i$.

有界对称域商上的交换子提升与插值

Commutant lifting and interpolation on quotients of bounded symmetric domains

研究有界对称域在有限复反射群作用下的商域上，Schur函数的交换子提升与插值问题，给出了压缩模映射存在Schur类提升的等价判据，并应用于有限点Nevanlinna-Pick型插值。

Commutant lifting and Nevanlinna-Pick interpolation in several variables

This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

Dilation theory, commutant lifting and semicrossed products

We take a new look at dilation theory for nonself-adjoint operator algebras. Among the extremal (co)extensions of a representation, there is a special property of being fully extremal. This allows a refinement of some of the classical notions which are important when one moves away from standard examples. We show that many algebras including graph algebras and tensor algebras of C*-correspondences have the semi-Dirichlet property which collapses these notions and explains why they have a better dilation theory. This leads to variations of the notions of commutant lifting and Ando's theorem. Th…

Interpolation without commutants

We introduce a "dual-space approach" to mixed Nevanlinna-Pick/Carathéodory-Schur interpolation in Banach spaces X of holomorphic functions on the disk. Our approach can be viewed as complementary to the well-known commutant lifting approach of D. Sarason and B. Nagy-C.Foiaş. We compute the norm of the minimal interpolant in X by a version of the Hahn-Banach theorem, which we use to extend functionals defined on a subspace of kernels without increasing their norm. This Functional extensions lemma plays a similar role as Sarason's Commutant lifting theorem but it only involves the predual of X a…

Hankel算子的提升与不变子空间

Liftings and invariant subspaces of Hankel operators

本文证明了Hankel变体的换位子提升定理，并利用核空间揭示了Hankel算子Beurling型约化与不变子空间的完整结构。

单位球上的逼近、插值与提升

Approximation, interpolation, and lifting on the unit ball

本文通过内函数和极值函数解决了单位球上的Nevanlinna-Pick插值问题，并推广了交换子提升定理，解决了多个著名问题。

All solutions to an operator Nevanlinna-Pick interpolation problem

The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna- Pick interpolation problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub-optimal Nehari problem and variation on the Nevanlinna-Pick interpolation problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorizat…