Commutators of small rank and reducibility of operator semigroups

Invariant subspaces for operator semigroups with commutators of rank at most one

Let X be a complex Banach space of dimension at least 2, and let S be a multiplicative semigroup of operators on X such that the rank of AB - BA is at most 1 for all pairs {A,B} in S. We prove that S has a non-trivial invariant subspace provided it is not commutative. As a consequence we show that S is triangularizable if it consists of polynomially compact operators. This generalizes results from [H. Radjavi, P. Rosenthal, From local to global triangularization, J. Funct. Anal. 147 (1997), 443-456] and [G. Cigler, R. Drnovšek, D. Kokol-Bukovšek, T. Laffey, M. Omladič, H. Radjavi, P. Rosenthal…

Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n.$ For any locally integrable function $b$, The commutators associated with $L^{-α/2}$ are defined by $[b,L^{-α/2}](f)(x)=b(x)L^{-α/2}(f)(x)-L^{-α/2}(bf)(x)$. When $b\in BMO(ω)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-α/2}]$ on weighted Morrey spaces respectively.

A Semigroup Composition C*-algebra

For 0 < s < 1, let phi_s(z)=sz+(1-s). We investigate the unital C*-algebra generated by the semigroup {C_{phi_s} : 0 < s < 1} of composition operators acting on the Hardy space of the unit disk. We determine the joint approximate point spectrum of a related collection of operators and show that the quotient of the C*-algebra by its commutator ideal is isomorphic to the direct sum of the complex numbers and the algebra of almost periodic functions on the real line. In addition, we show that the C*-algebra is irreducible.

Estimates for multilinear commutators of generalized fractional integral operators on weighted Morrey spaces

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(\mathbb{R}^n)$ with Gaussian kernel bounds, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n$. Assume that $\vec{b}=(b_1,b_2,\cdots,b_m)$ is a finite family of locally integrable functions, then the multilinear commutators generated by $\vec{b}$ and $L^{-α/2}$ is defined by \begin{equation*} L_{\vec{b}}^{-α/2}f=[b_m,\cdots,[b_2,[b_1,L^{-α/2}]],\cdots,]f \end{equation*} when $b_j\in BMO(w)$, $j=1,2,\cdots,m$, the authors obtain the boundedness of $L_{\vec{b}}^{-α/2}$ on weighted Morrey spaces.

Extremely Non-symmetric, Non-multiplicative, Non-commutative Operator Spaces

Motivated by importance of operator spaces contained in the set of all scalar multiples of isometries ($MI$-spaces) in a separable Hilbert space for $C^*$-algebras and E-semigroups we exhibit more properties of such spaces. For example, if an $MI$-space contains an isometry with shift part of finite multiplicity, then it is one-dimensional. We propose a simple model of a unilateral shift of arbitrary multiplicity and show that each separable subspace of a Hilbert space is the range of a shift. Also, we show that $MI$-spaces are non-symmetric, very unfriendly to multiplication, and prove a Comm…

无界算子分裂的一种初等方法

An elementary approach to splittings of unbounded operators

本文用初等方法推导了 Hilbert 空间中可能无界算子 $A$、$B$ 的 $e^{(A+B)}$ 三种常用分裂及其误差界，误差由交换子范数控制。