Some enclosures for the spectrum of J-frame operators

FRANCISCO MARTÍNEZ PERÍA

Cracovia

Conferencia; Spectral Theory and Applications (STA 2017); 2017

Faculty of Applied Mathematics, AGH University of Science and Technology

Recently, $J$-frame were introduced in order to extend frame-theory from Hilbert spaces to Krein spaces. Given aKrein space $\mathcal{H}$, a $J$-frame is a family of non-neutral vectors $\mathcal{F} =\{f_i\}_{i\in I}$which spans a pair of maximal uniformly definite subspaces. For each $J$-frame $\mathcal{F}$ there is an associatedoperator $S : \mathcal{H} \rightarrow \mathcal{H}$, selfadjoint with respect to the indefinite inner-product, which allows to reconstruct anyvector of the Krein space as a (possibly infinite) linear combination of the vectors in $\mathcal{F}$. Along this talk, we introduce $J$-frames [1], we briefly characterize $J$-frame operators and, finally, we present some spectral enclosures for a $J$-frame operator [2].This talk is based on joint works with Juan I. Giribet (IAM-UBA), Matthias Langer (U Strathclyde), Leslie Leben (TU Ilmenau), Alejandra Maestripieri (IAM-UBA), Pedro Massey (IAM-UNLP), Friedrich Philipp (KU Eichst$\ddot{\rm a}$tt-Ingolstadt), and Carsten Trunk (TU Ilmenau).