A Collection of Problems on Spectrally Bounded Operators

More elementary operators that are spectrally bounded

We discuss some necessary and some sufficient conditions for an elementary operator $x\mapsto\sum_{i=1}^n a_ixb_i$ on a Banach algebra $A$ to be spectrally bounded. In the case of length three, we obtain a complete characterisation when $A$ acts irreducibly on a Banach space of dimension greater than three.

1-bounded entropy and regularity problems in von Neumann algebras

We investigate the singular subspace of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N-N subbimodule of L^{2}(M) and it contains the quasinormalizer introduced by Popa, one-sided quasinormalizer introduced by Fang-Gao-Smith, and wq-normalizer introduced in Galatan-Popa (following upon work in Ioana-Peterson-Popa and Popa). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N in M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras, we show that there can be…

Operators with analytic orbit for the torus action

Let $T^n$ denote the n-dimensional torus. The class of the bounded operators on $L^2(T^n)$ with analytic orbit under the action of $T^n$ by conjugation with the translation operators is shown to coincide with the class of the zero-order pseudodifferential operators on $T^n$ whose discrete symbol $(a_j)_{j\in Z^n}$ is uniformly analytic, in the sense that there exists $C>1$ such that the derivatives of $a_j$ satisfy $|\partial^αa_j(x)|\leq C^{1+|α|}α!$ for all $x\in T^n$, all $j\in Z^n$ and all $α\in N^n$. This implies that this class of pseudodifferential operators is a spectrally invariant *-…

On the non-hypercyclicity of scalar type spectral operators and collections of their exponentials

Generalizing the case of a normal operator in a complex Hilbert space, we give a straightforward proof of the non-hypercyclicity of a (bounded or unbounded) scalar type spectral operator $A$ in a complex Banach space as well as of the collection $\left\{e^{tA}\right\}_{t\ge 0}$ of the exponentials of such an operator, which, under a certain condition on the spectrum of the operator $A$, coincides with the $C_0$-semigroup generated by $A$. The spectrum of $A$ lying on the imaginary axis, we also show that non-hypercyclic is the strongly continuous group $\left\{e^{tA}\right\}_{t\in {\mathbb R}}…

关于正规算子、其指数算子族及对称算子的非超循环性

On the non-hypercyclicity of normal operators, collections of their exponentials, and symmetric operators

该研究针对复希尔伯特空间中的正规算子、其指数算子族及对称算子，给出了它们非超循环性的简单直接证明，补充了对称算子非超循环性的相关结论。

保持Hilbert C*-模块的C*-子模映射

C*-submodule preserving module mappings on Hilbert C*-modules

研究Hilbert C*-模块上保持C*-子模的有界模映射的结构，证明其形式为d·id_X，其中d为可逆元素。