Invariant subspaces for Bishop operators and beyond

The Bishop-Phelps-Bollob{á}s property for numerical radius of operators on $L_1 (μ)$

In this paper, we introduce the notion of the Bishop-Phelps-Bollobás property for numerical radius (BPBp-$ν$) for a subclass of the space of bounded linear operators. Then, we show that certain subspaces of $\mathcal{L}(L_1(μ))$ have the BPBp-$ν$ for every finite measure $μ$. As a consequence we deduce that the subspaces of finite-rank operators, compact operators and weakly compact operators on $L_1(μ)$ have the BPBp-$ν$.

Bishop's Theorem and Differentiability of a subspace of $C_b(K)$

Let $K$ be a Hausdorff space and $C_b(K)$ be the Banach algebra of all complex bounded continuous functions on $K$. We study the Gâteaux and Fréchet differentiability of subspaces of $C_b(K)$. Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace $H$ of $C_b(K)$ is a dense $G_δ$ subset of $H$, if $K$ is compact. This gives a generalized Bishop's theorem, which says that the closure of the set of strong peak point for $H$ is the smallest closed norming subset of $H$. The classical Bishop's theorem was proved for a separating subalgebra $H$ a…

On the compact operators case of the Bishop-Phelps-Bollobás property for numerical radius

We study the Bishop-Phelps-Bollobás property for numerical radius restricted to the case of compact operators (BPBp-nu for compact operators in short). We show that $C_0(L)$ spaces have the BPBp-nu for compact operators for every Hausdorff topological locally compact space $L$. To this end, on the one hand, we provide some techniques allowing to pass the BPBp-nu for compact operators from subspaces to the whole space and, on the other hand, we prove some strong approximation property of $C_0(L)$ spaces and their duals. Besides, we also show that real Hilbert spaces and isometric preduals of $\…

Locating Ax, where A is a subspace of B(H)

Let A be a linear space of operators on a Hilbert space H, x a vector in H, and Ax the subspace of H comprising all vectors of the form Tx with T in A. We discuss, within a Bishop-style constructive framework, conditions under which the projection [Ax] of H on the closure of Ax exists. We derive a general result that leads directly to both the open mapping theorem and our main theorem on the existence of [Ax].

范围强暴露算子的Bollobás型定理

Bollobás-type theorems for range strongly exposing operators

本文研究了范围强暴露算子的Bollobás型定理，通过BPBp-RSE性质特征化了统一凸性和复统一凸性，并探讨了实数与复数情况下的差异及扩展结果。

次ordination算子的局部谱理论：Cesàro算子及其推广

Local spectral theory for subordinated operators: the Cesàro operator and beyond

本文研究由C0-半群产生的次ordination算子的局部谱性质，证明其缺乏SVEP但有稠密谱子空间，并应用于Cesàro算子及双曲复合半群，揭示其谱性质由Koenigs域几何决定。