Video library: O. I. Reinov, The symmetry of a spectrum of nuclear operators in subspaces of $L

VIDEO LIBRARY


International conference on Function Spaces and Approximation Theory dedicated to the 110th anniversary of S. M. Nikol'skii May 28, 2015 17:30, Moscow, Steklov Mathematical Institute of RAS

The symmetry of a spectrum of nuclear operators in subspaces of $L_p$-spaces O. I. Reinov Saint Petersburg State University
Abstract: It was proved in the paper [1] that the spectrum of a nuclear operator $A$ acting on a separable Hilbert space is central-symmetric if and only if $\operatorname{trace}A^{2n - 1} = 0$, $n \in \mathbb N$. We prove: Theorem. Let $Y$ be a subspace of a quotient (or a quotient of a subspace) of an $L_p$-space, $1\le p\le\infty$ and $T\in N_s(Y,Y)$ ($s$-nuclear), where $1/s=1+\|1/2-1/p\|$. The spectrum of $T$ is central-symmetric if and only if $\operatorname{trace}A^{2n - 1} = 0$, $n =1,2,\dots$ . Remark. $T$ is $s$-nuclear, if $T$ admits a representation $$ T=\sum_i \lambda_i y'_i\otimes y_i, $$ where $(\lambda_i)\in l_s,$ $(y'_i)$ and $(y_i)$ are bounded. Language: English References M. I. Zelikin, “A criterion for the symmetry of a spectrum”, Dokl. Akad. Nauk, 418:6 (2008), 737–740