arXiv subjects

Elif Uyanık

Publications and source records attributed to Elif Uyanık.

4 recordsLinked to original sources

arXiv2017

A note on triangular operators on Smooth Sequence Spaces

For a scalar sequence {(θ_n)}_{n \in \mathbb{N}}, let C be the matrix defined by c_n^k = θ_{n-k+1} if n > k, c_n^k = 0 if n < k. The map between Köthe spaces λ(A) and λ(B) is called a Cauchy Product map if it is determined by the triangular matrix C. In this note we introduced some necessary and sufficient conditions for a Cauchy Product map on a nuclear Köthe space λ(A) to nuclear G_1-space λ(B) to be linear and continuous. Its transpose is also considered.

math.FA↗

arXiv2017

Bounded Operators to $\ell$-Köthe Spaces

For Fr{é}chet spaces E and F we write (E,F) \in {B} if every continuous linear operator from E to F is bounded. Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We obtain a necessary and sufficient condition for (E,F) \in {B} when F = λ^{l}(B). We say that a triple (E,F,G) has the bounded factorization property and write (E,F,G) \in {BF} if each continuous linear operator T : E \longrightarrow G that factors over F is bounded. We extend some results in \cite{Ter03} to l-Köthe spaces and obtain a sufficient condition for (E,λ^{l_1}(A) \hat{otimes}_{pi} λ^{l_2}(B), λ^{l_3}(C)) \in {BF} when λ^{l_1}(A) and λ^{l_2}(B) are nuclear.

math.FA↗

arXiv2017

A Remark on a paper of P. B. Djakov and M. S. Ramanujan

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-Köthe spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding Köthe matrices when every continuous linear operator between l-Köthe spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-Köthe spaces, under a splitting condition, causes the existence of a common basic subspace.

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↗