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Ersin Kızgut

Publications and source records attributed to Ersin Kızgut.

7 recordsLinked to original sources
arXiv2021

Some stability results in projective tensor products

Let $(E,F)$ be a pair of Fréchet spaces. In this paper, we discuss whether a certain property $P$ enjoyed by both $E$ and $F$ is also satisfied by the complete tensor product $E \widehat{\otimes}_π F$. Specifically we focus on the two properties generalizing Köthe spaces: the normability condition called the property $(y)$, and the property of smallness of bounded subsets of Fréchet spaces up to a complemented Banach subspace in connection with the problem of topologies of A. Grothendieck. We also consider the stability of some well known equivalences of operator classes with an application in the problem whether the sum of complemented subspaces is also complemented.

math.FA↗
arXiv2020

The Cesàro operator on weighted Bergman Fréchet and (LB)-spaces of analytic functions

The spectrum of the Cesàro operator $\mathsf{C}$ is determined on the spaces which arises as intersections $A^p_{α+}$ (resp. unions $A^p_{α-}$) of Bergman spaces $A_α^p$ of order $1<p<\infty$ induced by standard radial weights $(1-|z|)^α$, for $0<α<\infty$. We treat them as reduced projective limits (resp. inductive limits) of weighted Bergman spaces $A^p_α$, with respect to $α$. Proving that these spaces admit the monomials as a Schauder basis paves the way for using Grothendieck-Pietsch criterion to deduce that we end up with a non-nuclear Fréchet-Schwartz space (resp. a non-nuclear (DFS)-space). We show that $\mathsf{C}$ is always continuous, while it fails to be compact or to have bounded inverse on $A^p_{α+}$ and $A^p_{α-}$.

math.FA↗
arXiv2018

The Cesàro operator on smooth sequence spaces of finite type

The discrete Cesàro operator $\mathsf{C}$ is investigated in the class of smooth sequence spaces $λ_0(A)$ of finite type. This class contains properly the power series spaces of finite type. Of main interest is its spectrum, which is distinctly different in the cases when $λ_0(A)$ is nuclear and when it is not. The nuclearity of $λ_0(A)$ is characterized via certain properties of the spectrum of $\mathsf{C}$. Moreover, $\mathsf{C}$ is always power bounded and uniformly mean ergodic on $λ_0(A)$.

math.FA↗
arXiv2017

Remarks on strictly singular operators

A continuous linear operator $T:E \to F$ is called strictly singular if it cannot be invertible on any infinite dimensional closed subspace of its domain. In this note we discuss sufficient conditions and consequences of the phenomenon $LB(E,F)=L_s(E,F)$, which means that every continuous linear bounded operator defined on $E$ into $F$ is strictly singular.

math.FA↗
arXiv2016

On the existence of a factorized unbounded operator between Fréchet spaces

For locally convex spaces $X$ and $Y$, the continuous linear map $T:X \to Y$ is called bounded if there is a zero neighborhood $U$ of $X$ such that $T(U)$ is bounded in $Y$. Our main result is that the existence of an unbounded operator $T$ between Fréchet spaces $E$ and $F$ which factors through a third Fréchet space $G$ ends up with the fact that the triple $(E, G, F)$ has an infinite dimensional closed common nuclear Köthe subspace, provided that $F$ has the property $(y)$.

math.FA↗
arXiv2016

Remarks on bounded operators in $\ell$-Köthe spaces

For locally convex spaces $X$ and $Y$, the continuous linear map $T:X \to Y$ is said to be bounded if it maps zero neighborhoods of $X$ into bounded sets of $Y$. We denote $(X,Y) \in \mathcal{B}$ when every operator between $X$ and $Y$ is bounded. For a Banach space $\ell$ with a monotone norm $\|\cdot\|$ in which the canonical system $(e_n)$ forms an unconditional basis, we consider $\ell$-Köthe spaces as a generalization of usual Köthe spaces. In this note, we characterize $\ell$-Köthe spaces $\ell(a_{pn})$ and $\ell(b_{sm})$ such that $(\ell(a_{pn}), \ell(b_{sm})) \in \mathcal{B}$. A pair $(X,Y)$ is said to have the bounded factorization property, and denoted $(X,Y) \in \mathcal{BF}$, if each linear continuous operator $T : X \to X$ that factors over $Y$ is bounded. We also prove that injective tensor products of some classical Köthe spaces have bounded factorization property.

math.FA↗