Related papers: Ces\`{a}ro Operators on Rooted Directed Trees

In a recent paper [9], R. E. Curto, S. H. Lee and J. Yoon asked the following question: Let $T$ be a subnormal operator, and assume that $T^2$ is quasinormal. Does it follow that $T$ is quasinormal?. In [36] we answered this question in the…

The Cauchy dual subnormality problem asks whether the Cauchy dual operator $T^{\prime}:=T(T^*T)^{-1}$ of a $2$-isometry $T$ is subnormal. In the present paper we show that the problem has a negative solution. The first counterexample…

We investigate the $n$th root problem for bounded operators on a Hilbert space within the class of conditionally positive definite (CPD) operators determined by the L\'evy--Khintchine formula. The class contains subnormal operators,…

We discuss the Ces`aro operator on the Hardy space in the upper half-plane. We provide a new simple proof of the boundedness of this operator, prove that this operator is equal to the sum of the identity operator and a unitary operator,…

We study local spectral properties for subordinated operators arising from $C_0$-semigroups. Specifically, if $\mathcal{T}=(T_t)_{t\geq 0}$ is a $C_0$-semigroup acting boundedly on a complex Banach space and $$\mathcal{H}_\nu =…

The main goal of this paper is to characterise all the possible Ces\`aro and $L$-asymptotic limits of powerbounded, complex matrices. The investigation of the $L$-asymptotic limit of a powerbounded operator goes back to Sz.-Nagy and it…

In this paper we study boundedness and detailed spectral properties for the Ces\`aro-Hardy operator and some generalizations in $L^p[0,1]$. The study employs $C_0$-semigroup theory, expressing the Ces\`aro-Hardy operators and their dual…

In this paper we prove that the Scott topology $\mathfrak S$ on a rooted non-metric tree $\mathcal T$ is strictly coarser than the weak tree topology. Moreover, for each $t\in \mathcal T$, we consider a natural order $\preceq_t$ on…

The discrete Ces\`aro operator $\mathsf{C}$ is investigated in strong duals of smooth sequence spaces of infinite type. Of main interest is its spectrum, which turns out to be distinctly different in the cases when the space is nuclear and…

We study several notions of boundedness for operators. It is known that any power bounded operator is absolutely Ces\`aro bounded and strong Kreiss bounded (in particular, uniformly Kreiss bounded). The converses do not hold in general. In…

In this paper, we extend the concept of absolutely Ces\`aro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if $T$ is an absolutely Ces\`{a}ro bounded…

The generalized Ces\`aro operators $C_t$, for $t\in [0,1]$, were first investigated in the 1980's. They act continuously in many classical Banach sequence spaces contained in $\mathbb{C}^{\mathbb{N}_0}$, such as $\ell^p$, $c_0$, $c$,…

Unlike for $\ell_p$, $1<p\leq\infty$, the discrete Ces\`aro operator $C$ does not map $\ell_1$ into itself. We identify precisely those weights $w$ such that $C$ does map $\ell_1(w)$ continuously into itself. For these weights a complete…

A metric tree is a tree whose edges are viewed as line segments of positive length. The Dirac operator on such tree is the operator which operates on each edge, complemented by the matching conditions at the vertices which were given by…

We study a class of rooted trees with a substitution type structure. These trees are not necessarily regular, but exhibit a lot of symmetries. We consider nearest neighbor operators which reflect the symmetries of the trees. The spectrum of…

The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any…

We study the asymptotic behaviour of orbits $(T^nx)_{n\ge0}$ of the classical Ces\`aro operator $T$ for sequences $x$ in the Banach space $c$ of convergent sequences. We give new non-probabilistic proofs, based on the Katznelson-Tzafriri…

In this article we study the Ces\`{a}ro operator $$ \mathcal{C}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\zeta)\,d\zeta, $$ and its companion operator $\mathcal{T}$ on Hardy spaces of the upper half plane. We identify $\mathcal{C}$ and $\mathcal{T}$…

The spectrum of the Ces\`aro operator $\mathsf{C}$ is determined on the spaces which arises as intersections $A^p_{\alpha +}$ (resp. unions $A^p_{\alpha -}$) of Bergman spaces $A_\alpha^p$ of order $1<p<\infty$ induced by standard radial…

For $\beta>0$ and $p\ge 1$, the generalized Ces\`aro operator $$ \mathcal{C}_\beta f(t):=\frac{\beta}{t^\beta}\int_0^t (t-s)^{\beta-1}f(s)ds $$ and its companion operator $\mathcal{C}_\beta^*$ defined on Sobolev spaces…