Xiaomei Cai | Searcharxiv

arXiv2018

The absolute values and support projections for a class of operator matrices involving idempotents

Let $λ\in \mathbb{R},$ $μ\in \mathbb{R}$ and $B$ be a linear bounded operator from a Hilbert space $\mathcal{K}$ into another Hilbert space $\mathcal{H}.$ In this paper, we consider the formulas of the absolute value $|Q_{λ,μ}|,$ where $Q_{λ,μ}$ with respect to the decomposition $\mathcal{H}\oplus\mathcal{K}$ have the operator matrix form $Q_{λ,μ}:=\left(\begin{array}{cc}λI&B\\B^*&μI\end{array}\right).$ Then the positive part and the support projection of $Q_{λ,0}$ are obtained. Also, we characterize the symmetry $J$ such that a projection $E$ is the $J$-projection. In particular, the minimal element of the set of all symmetries $J$ with the property $JE\geqslant0$ is described.

math.FA↗

arXiv2018

The minimal and maximal symmetries for $J$-contractive projections

In this paper, we firstly character the structures of symmetries $J$ such that a projection $P$ is $J$-contractive. Then the minimal and maximal elements of the symmetries $J$ with $P^{\ast}JP\leqslant J$(or $JP\geqslant0)$ are given. Moreover, some formulas between $P_{(2I-P-P^{\ast})^{+}}$ $(P_{(2I-P-P^{\ast})^{-}})$ and $P_{(P+P^{\ast})^-}$ $(P_{(P+P^{\ast})^+})$ are established.

math.FA↗