Near Normal Dilations of Nonnormal Matrices and Linear Operators

Abstract

Let A be a square matrix or a linear operator on a Hilbert space H. A dilation of A is a linear operator M on a larger space K ⊃ H such that A = P H M | H , where P H is orthogonal projection onto 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.A reason for studying dilations is that the operator M may have some nice properties that A does not have. For example, if A is highly nonnormal, then it is wellknown that the spectrum σ(A) may give little information about the 2-norm behavior of functions of A. If M is a normal or near normal power dilation of A, however, then p(A) ≤ p(M ) for any polynomial p (since p(A) is a block of p(M )), and if · denotes the 2-norm (as it will throughout this paper), then p(M ) is determined or approximately determined by σ(M ). That is, if M is similar to a normal operator N via a similarity transformation with a moderate condition number Q (M

Dynamical Deformation of Toroidal Matrix Varieties

“…The approximation and connectivity technology developed in this paper has a natural connection to approximate simultaneous diagonalization of m-tuples of matrices (in the sense of [24]) and to normal matrix approximation of almost normal matrices in the sense of [12,16]. The development of numerical algorithms to perform these tasks will be the subject of future communications.…”

“…The application and extension of the results in §3 to the study of normal and near normal compressions of normal matrices (in the sense of [12,14] and [1, §9 and §10]) will be the subject of further study as well.…”

“…In this study we will use the C * -algebraic technology developed in [10,21,22,29] to derive a connectivity technique that can be used to study and solve problems that appear in clustered matrix approximation (in the sense of [28]) and matrix dilation/compression problems (in the sense of [12,14] and [1, §9 and §10]).…”