Decay properties for functions of matrices overC⁎-algebras

Abstract

International audienceWe extend previous results on the exponential off-diagonal decay of the entries of analytic functions of banded and sparse matrices to the case where the matrix entries are elements of a C⁎-algebra

Decay Bounds for Functions of Hermitian Matrices with Banded or Kronecker Structure

“…This result was extended in [7] to the case where M is a sparse matrix with a general sparsity pattern, using the graph distance instead of the distance from the main diagonal; see also [14,33] and section 5 below. Similar bounds for analytic functions of non-Hermitian matrices can be found in [4,7].…”

“…The study of the decay behavior for general analytic functions of banded matrices, including the important case of the matrix exponential, was initiated in [6,32] and continued for possibly nonnormal matrices and general sparsity patterns in [7]; further contributions in these directions include [4,16,38,42]. Collectively, these papers have largely elucidated the question of when one can expect exponential decay in the entries of f (A), in terms of conditions that the function f and the matrix A must satisfy.…”

“…A related but quite distinct line of research concerned the study of inverse-closed matrix algebras, where the decay behavior in the entries of a (usually infinite) matrix A is "inherited" by the entries of A −1 . Here we mention [33], where it was observed that a similar decay behavior occurs for the entries of f (A) = A −1/2 , as well as [2,3,26,27,35], among others.The study of the decay behavior for general analytic functions of banded matrices, including the important case of the matrix exponential, was initiated in [6,32] and continued for possibly nonnormal matrices and general sparsity patterns in [7]; further contributions in these directions include [4,16,38,42]. Collectively, these papers have largely elucidated the question of when one can expect exponential decay in the entries…”

Bounds for the decay of the entries in inverses and Cauchy–Stieltjes functions of certain sparse, normal matrices

“…This class of matrices includes (shifted) skew‐Hermitian and (shifted) Hermitian indefinite matrices as important special cases. Results for general normal matrices can be found in previous works . While these can be applied to general analytic functions, as long as f ( A ) is defined, most of the results fail to be insightful in practical situations, as they typically depend on quantities that are very hard or impossible to compute in practice (e.g., because they require knowledge of the complete spectrum of the matrix A ) or are very pessimistic and therefore do not capture the actual quantitative decay behavior well.…”

“…Results for general normal matrices can be found in previous works. 11,13,14,19 While these can be applied to general analytic functions, as long as f(A) is defined, most of the results fail to be insightful in practical situations, as they typically depend on quantities that are very hard or impossible to compute in practice (e.g., because they require knowledge of the complete spectrum of the matrix A) or are very pessimistic and therefore do not capture the actual quantitative decay behavior well. Recently, 12 sharp and practicable decay bounds were obtained for functions that are analytic on a connected set containing the field of values of A.…”