Solution of a problem of Peller concerning similarity

Peller's problem concerning Koplienko-Neidhardt trace formulae: the unitary case

We prove the existence of a complex valued $C^2$-function on the unit circle, a unitary operator U and a self-adjoint operator Z in the Hilbert-Schmidt class $S^2$, such that the perturbated operator $$ f(e^{iZ}U)-f(U) -\frac{d}{dt}\bigl(f(e^{itZ}U)\bigr)_{\vert t=0} $$ does not belong to the space $S^1$ of trace class operators. This resolves a problem of Peller concerning the validity of the Koplienko-Neidhardt trace formula for unitaries.

Resolution of Peller's problem concerning Koplienko-Neidhardt trace formulae

A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert-Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty$ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko-Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \…

Joint similarity to operators in noncommutative varieties

In this paper we solve several problems concerning joint similarity to n-tuples of operators in noncommutative varieties in $[B(\cH)^n]_1$ associated with positive regular free holomorphic functions in $n$ noncommuting variables and with sets of noncommutative polynomials in $n$ indeterminates, where $B(\cH)$ is the algebra of all bounded linear operators on a Hilbert space $\cH$. In particular, if $f=X_1+...+X_n$ and $\cP=\{0\}$, the elements of the corresponding variety can be seen as noncommutative multivariable analogues of Agler's $m$-hypercontractions.

Similarity problems, Folner sets and isometric representations of amenable semigroups

We revisit Sz.-Nagy's criteria for similarity of Hilbert space bounded linear operators to isometries or unitaries and present new ones. We also discuss counterparts of the Dixmier-Day theorem concerning bounded representations of amenable groups and semigroups. We highlight the role of Folner sets in similarity problems in both settings of unimodular, $σ$-compact, amenable groups and in discrete semigroups possessing the Strong Folner condition (SFC).

非交换加权移位、联合相似性与多变量函数论

Noncommutative weighted shifts, joint similarity, and function theory in several variables

本文研究非交换加权移位的结构性质，得到n元算子组与非交换加权多移位的Rota型联合相似结果，构造拟幂零算子的非交换多变量Foias-Pearcy模型，并发展非交换形式幂级数理论与算子函数演算。

希尔伯特空间算子的稳定性关系与卡普兰斯基问题

Stability relations for Hilbert space operators and a problem of Kaplansky

本文研究卡普兰斯基第二问题的算子版本，在加强假设下证明紧算子情形下相似性成立，并给出反例及不可数酉算子族。