Narrow operators are those operators defined on function spaces which are "small" at signs, i.e., at {-1,0,1}-valued functions. We summarize here some results and problems on them. One of the most interesting things is that if E has an unconditional basis then each operator on E is a sum of two narrow operators, while the sum of two narrow operators on L₁ is narrow. Recently this notion was generalized to vector lattices. This generalization explained the phenomena of sums: the set of all regular...

Let X be a Banach space. We introduce a formal approach which seems to be useful in the study of those properties of operators on X which depend only on the norms of the images of elements. This approach is applied to the Daugavet equation for norms of operators; in particular we develop a general theory of narrow operators and rich subspaces of spaces X with the Daugavet property previously studied in the context of the classical spaces C(K) and L₁(μ).

The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow...

The Nevanlinna algebras, ${}_{\alpha }^p$, of this paper are the $L^p$ variants of classical weighted area Nevanlinna classes of analytic functions on = z ∈ ℂ: |z| < 1. They are F-algebras, neither locally bounded nor locally convex, with a rich duality structure. For s = (α+2)/p, the algebra $F_s$ of analytic functions f: → ℂ such that ${(1-|z\left|\right)}^s\left|f\left(z\right)\right|\to 0$ as |z| → 1 is the Fréchet envelope of ${}_{\alpha }^p$. The corresponding algebra ${}_s^{\infty }$ of analytic f: → ℂ such that $su{p}_{z\in }{(1-|z\left|\right)}^s\left|f\left(z\right)\right|<\infty$ is a complete metric space but fails to be a topological vector space. $F_s$ is also...

Let $u$ be a holomorphic function and $\varphi$ a holomorphic self-map of the open unit disk $𝔻$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $u{C}_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $𝔻$ in terms of $u$, $\varphi$, their derivatives, and ${\varphi }^n$, the $n$-th power of $\varphi$. Moreover, we obtain some similar estimates for the essential norms of the operators $u{C}_{\varphi }$, from which sufficient and necessary conditions of compactness of $u{C}_{\varphi }$ follows immediately.

Let ψ and φ be analytic functions on the open unit disk $𝔻$ with φ($𝔻$) ⊆ $𝔻$. We give new characterizations of the bounded and compact weighted composition operators W ψ,ϕ from the Hardy spaces H p, 1 ≤ p ≤ ∞, the Bloch space B, the weighted Bergman spaces A αp, α > − 1,1 ≤ p < ∞, and the Dirichlet space $𝒟$ to the Bloch space in terms of boundedness (respectively, convergence to 0) of the Bloch norms of W ψ,ϕ f for suitable collections of functions f in the respective spaces. We also obtain characterizations...

Let $X$ be a Banach space of analytic functions on the open unit disk and $\Gamma$ a subset of linear isometries on $X$. Sufficient conditions are given for non-supercyclicity of $\Gamma$. In particular, we show that the semigroup of linear isometries on the spaces $S^p$ ($p>1$), the little Bloch space, and the group of surjective linear isometries on the big Bloch space are not supercyclic. Also, we observe that the groups of all surjective linear isometries on the Hardy space $H^p$ or the Bergman space $L_a^p$ ($1<p<\infty$, $p\ne 2$) are not supercyclic....

We consider operators of the form $\left(\Omega f\right)\left(y\right)={ʃ}_{-\infty }^{\infty }\Omega (y,u)f\left(u\right)du$ with Ω(y,u) = K(y,u)h(y-u), where K is a Calderón-Zygmund kernel and $h\in L^{\infty }$ (see (0.1) and (0.2)). We give necessary and sufficient conditions for such operators to map the Besov space ${Ḃ}_1^{0,1}$ (= B) into itself. In particular, all operators with $h\left(y\right)=e^{{i\left|y\right|}^a}$, a > 0, a ≠ 1, map B into itself.

We show that a positive semigroup $T_t$ on $L_p(\Omega ,\nu )$ with generator A and ||R(α + i β)|| → 0 as |β| → ∞ for some α ∈ ℝ is continuous in the operator norm for t>0. The proof is based on a criterion for norm continuity in terms of “smoothing properties” of certain convolution operators on general Banach spaces and an extrapolation result for the $L_p$-scale, which may be of independent interest.

We show that the numerical index of a $c_0$-, $l_1$-, or $l_{\infty }$-sum of Banach spaces is the infimum of the numerical indices of the summands. Moreover, we prove that the spaces C(K,X) and $L_1(\mu ,X)$ (K any compact Hausdorff space, μ any positive measure) have the same numerical index as the Banach space X. We also observe that these spaces have the so-called Daugavet property whenever X has the Daugavet property.