Equations $AX=Y$ and $Ax=y$ in Alg$\Cal L$

Abstract

Let $\Cal L$ be a subspace lattice on a Hilbert space $\Cal H$ and $X$ and $Y$ be operators acting on a Hilbert space ${\Cal H}$. Let $P$ be the projection onto $\overline{{\Cal R} (X)}$, where ${\Cal R} X$ is the range of $X$. If $PE=EP$ for each $E\in {\Cal L}$, then there exists an operator $A$ in Alg$\Cal L$ such that $AX=Y$ if and only if $$\displaystyle\sup\{ {{\|E^\bot Yf\|} / {\|E^\bot Xf\|}} : f\in {\Cal H}, ~E\in {\Cal L} \}= K <\infty.$$ Moreover, if the necessary condition holds, then we may choose an operator $A$ such that $AX=Y$ and $\|A\| = K$. Let $x$ and $y$ be vectors in ${\Cal H}$ and let $P_x$ be the projection onto the singlely generated space by $x$. If $P_xE=EP_x$ for each $E\in {\Cal L}$, then the assertion that there exists an operator $A$ in Alg$\Cal L$ such that $Ax=y$ is equivalent to the condition $$K_0 : = \displaystyle\sup\{ {{\|E^\bot y\|} / {\|E^\bot x\|}} : ~E\in {\Cal L} \} <\infty.$$ Moreover, we may choose an operator $A$ such that $\|A\|=K_0$ whose norm is $K_0 $ under this case.