Closed densely defined operators commuting with multiplications in a multiplier pair

Abstract:

For a multiplier pair $\left ( X,Y\right )$ we study the closed densely defined operators $T$ on $X$ that commute with all of the multiplications by right multipliers in $X$. We apply our general results to special cases involving $H^{p}$, completions of $L^{\infty }\left [ 0,1\right ]$ with respect to certain norms, and the completion of a $II_{1}$ factor von Neumann algebra with respect to a unitarily invariant norm, where we show that each such $T$ is a “left multiplication”. However, we give an example of a closed densely defined operator on the Bergman space that commutes with all multiplications by $H^{\infty }$-functions but is not a multiplication operator.