arXiv subjects
Publications and source records attributed to Amir Bahman Nasseri.
Let $R$ be the closed linear span of the Rademacher functions in $L^1(0,1)$. We prove that $L^1(0,1)/R$ embeds isomorphically into $L^1(0,1)$, answering affirmatively a question of González and Martínez-Abejón \cite[p.~140]{GM97}. We then use this embedding to show that the weak Calkin algebra of the quotient is uniformly incompressible. \end{abstract
We show that the norm-closed linear span of products of two Dunford-Pettis operators on $L^1(0,1)$ lies strictly between the representable and the Dunford-Pettis operators. This gives an additional proper large closed ideal in $\mathcal{B}(L^1(0,1))$, answering the existence question of Johnson, Pisier and Schechtman. The quotient of the Dunford-Pettis ideal by this square ideal contains a contractively complemented isometric copy of $L^1(0,1)$. We also prove that the Dunford-Pettis ideal has no right approximate identity, answering a question of Johnson and Schechtman. More precisely, a positive norm-one convolution operator $R$ satisfies $\|R-RT\|\geq1$ and $\|R-TR\|\geq1$ for every Dunford-Pettis operator $T$. The proofs combine translation averaging, Rajchman measures supported on a strongly independent set, and two atomless disintegrations of positive operators. The approximate-identity obstruction extends to every finite measure space with a nonzero atomless part. Finally, we prove that quantitative multiplication estimates for an ideal pass to all its closed power ideals.
We study the representation of non-weakly compact operators between $AL$-spaces. In this setting, we show that every operator admits a best approximant in the ideal of weakly compact operators. Using duality arguments, we extend this result to operators between $C(L)$-spaces where $L$ is extremally disconnected. We also characterize the weak essential norm for operators between $AL$-spaces in terms of factorizations of the identity on $\ell_1$. As a consequence, we deduce that the weak Calkin algebra $\mathscr{B}(E)/\mathscr{W}(E)$ admits a unique algebra norm for every $AL$-space $E$. By duality, similar results are obtained for $C(K)$-spaces. In particular, we prove that for operators $T: L_{\infty}[0,1] \to L_{\infty}[0,1]$ the weak essential norm, the residuum norm, and the De Blasi measure of weak compactness coincide, answering a question of González, Saksman and Tylli.
There exist injective Tauberian operators on $L_1(0,1)$ that have dense, non closed range. This gives injective, non surjective operators on $\ell_\infty$ that have dense range. Consequently, there are two quasi-complementary, non complementary subspaces of $\ell_\infty$ that are isometric to $\ell_\infty$.
This article deals with properties of spectra of operators on C(K)-spaces with the Grothendieck property (e.g. l^{\infty}) and application to so called J-class operators introduced by A. Manoussos and G. Costakis. We will show that C(K) has the Grothendieck property if and only if the boundary of the spectrum of every operator on C(K) consists entirely of eigenvalues of its adjoint. As a consequence we will see that there does not exist invertible J-class operators on C(K) with the Grothendieck property. In the third section we will give a quantitative and qualitative characterization of all J-class operators on l^{\infty} which are adjoints from operators on l^1.
Let $B$, $I$ be the unweighted backward shift and the identity operator respectively on $l^{\infty}(\mathbb{N})$, the space of bounded sequences over the complex numbers endowed with the supremum norm. We prove that $I+λB$ is locally topologically transitive if and only if $|λ|>2$. This, shows that a classical result of Salas, which says that backward shift perturbations of the identity operator are always hypercyclic, or equivalently topologically transitive, on $l^p(\mathbb{N})$, $1\leq p<+\infty$, fails to hold for the notion of local topological transitivity on $l^{\infty}(\mathbb{N})$. We also obtain further results which complement certain results from \cite{CosMa}.