Near Normal Dilations of Nonnormal Matrices and Linear Operators
Anne Greenbaum, Trevor Caldwell, Kenan Li · SIAM Journal on Matrix Analysis and Applications · 2016
Let $A$ be a square matrix or a linear operator on a Hilbert space ${\cal H}$. A dilation of $A$ is a linear operator $M$ on a larger space ${\cal K} \supset {\cal H}$ such that $A = \left. P_{{\cal H}} M \right|_{\cal H}$, where $P_{{\cal H}}$ is orthogonal projection onto ${\cal H}$. Often it is required additionally that $M^k$ be a dilation of $A^k$ for all or a range of positive integer powers $k$. While much work has been aimed at proving existence of dilations with various properties, there has been little study of the behavior of functions of these dilations and how it compares to that of the original operator. Is the original operator a major part of the dilated one or is it an insignificant piece? Does the larger operator represent some physical process, where the original operator might be an important component for certain times but not for others? In this paper we construct near normal dilations of nonnormal matrices, with the spectrum of the dilated operator around the boundary of the numerical range of the matrix. We compare the behavior of $e^{tA}$ and $e^{tM}$ for $t > 0$. We find that the dilated operator takes on a life of its own, representing a wave that grows or decays but eventually dominates the part corresponding to the original operator. We derive other near normal dilations in which this behavior is less pronounced.