Property $(\omega)$ for 2-by-2 upper-triangular operator matrices
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With given parameters $A\in\mathcal{B(H)}$ and $B\in\mathcal{B(K)}$, we denote by $M_C$ an operator of the following type $M_C:=\left(\begin{array}{cc} A & C \\ 0 & B \end{array} \right)$, acting on the Hilbert space $\mathcal{H\oplus K}$. First, this paper gives necessary and sufficient conditions for $M_C$ to obey property $(\omega)$. More precisely, the conditions on its diagonal operators $A$, $B$ and skew-diagonal operator matrix $M_0:=\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array} \right)$ ensuring property $(\omega)$ for $M_C$ have been investigated. Moreover, we explore how property $(\omega)$ survives for functions of operator matrices.
Abstract
With given parameters $A\in\mathcal{B(H)}$ and $B\in\mathcal{B(K)}$, we denote by $M_C$ an operator of the following type $M_C:=\left(\begin{array}{cc} A & C \\ 0 & B \end{array} \right)$, acting on the Hilbert space $\mathcal{H\oplus K}$. First, this paper gives necessary and sufficient conditions for $M_C$ to obey property $(\omega)$. More precisely, the conditions on its diagonal operators $A$, $B$ and skew-diagonal operator matrix $M_0:=\left(\begin{array}{cc} A & 0 \\ 0 & B \end{array} \right)$ ensuring property $(\omega)$ for $M_C$ have been investigated. Moreover, we explore how property $(\omega)$ survives for functions of operator matrices.