On some perturbation inequalities for operators (original) (raw)
Perturbation Theory for Linear Operators Springer
summary. Chapters are divided into sections, and sections into paragraphs. I- § 2.3, for example, means paragraph three of section two of chapter one; it is simply written § 2.3 when referred to within the same chapter and par. 3 when referred to within the same section. Theorems, Corollaries, Lemmas, Remarks, Problems, and Examples are numbered in one list within each section: Theorem 2.1, Corollary 2.2, Lemma 2.3, etc. Lemma 1-2.3 means Lemma 2.3 of chapter one, and it is referred to simply as Lemma 2.3 within the same chapter. Formulas are numbered consecutively within each section; I-(2.3) means the third formula of section two of chapter one, and it is referred to as (2.3) within the same chapter. Some of the problems are disguised theorems, and are quoted in later parts of the book. Numbers in [ ] refer to the first part of the bibliography containing articles, and those in Q JI to the second part containing books and monographs. There are a subject index, an author index and a notation index at the end of the book. The book was begun when I was at the University of Tokyo and completed at the University of California. The preparation of the book has been facilitated by various financial aids which enabled me to pursue research at home and other institutions. For these aids I am grateful to the following agencies: the Ministry of Education, Japan; Commissariat General du Plan, France; National Science Foundation,
Perturbations of Operator Functions in a Hilbert Space
2012
Let A andà be linear bounded operators in a separable Hilbert space, and f be a function analytic on the closed convex hull of the spectra of A andÃ. Let S N 2 and S N 1 be the ideals of Hilbert-Schmidt and nuclear operators, respectively. In the paper, a sharp estimate for the norm of f (A) − f (Ã) is established, provided A andà have the so called Hilbert-Schmidt property. In particular, A has the Hilbert-Schmidt property, if one of the following conditions holds:
Functions of perturbed normal operators
2010
In [10], [11], [1], [2], and sharp estimates for f (A) − f (B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this note we extend those results to the case of functions of normal operators. We show that if f belongs to the Hölder class Λα(R 2 ), 0 < α < 1, of functions of two variables, and N 1 and N 2 are normal operators, then f (N 1 ) − f (N 2 ) ≤ const f Λα N 1 − N 2 α . We obtain a more general result for functions in the space Λω(R 2 ) = f : |f (ζ 1 ) − f (ζ 2 )| ≤ const ω(|ζ 1 − ζ 2 |) for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class B 1 ∞1 (R 2 ), then it is operator Lipschitz, i.e.,
Estimates for norms of matrix-valued and operator-valued functions and some of their applications
Acta Applicandae Mathematicae, 1993
AbstracL A survey is presented of estimates for a norm of matrix-valued and operator-valued functions obtained by the author. These estimates improve the Gel'fand-Shilov estimate for regular functions of matrices and Carleman's estimates for resolvents of matrices and compact operators. From the estimates for resolvents, the well-known result for spectrum perturbations of self-adjoint operators is extended to quasi-Hermitian operators. In addition, the classical Schur and Brown's inequalities for eigenvalues of matrices are improved. From estimates for the exponential function (semigroups), bounds for solution norms of nonlinear differential equations are derived. These bounds give the stability criteria which make it possible to avoid the construction of Lyapunov functions in appropriate situations.
Perturbation of closed range operators
Turk J Math, 2009
Let T, A be operators with domains D (T)⊆ D (A) in a normed space X. The operator A is called T-bounded if Ax≤ ax+bTx for some a, b≥ 0 and all x∈ D (T). If A has the HyersUlam stability then under some suitable assumptions we show that both T and S:= A+T ...
Estimates in the Operator Norm
In this paper, we will obtain estimates of the distance between the q-k-eigenvalues of two q-k-normal matrices A and B interms of AB . Apart from the optimal matching distances.
Perturbation theory for normal operators
Transactions of the American Mathematical Society, 2013
Let E ∋ x → A(x) be a C -mapping with values unbounded normal operators with common domain of definition and compact resolvent. Here C stands for C ∞ , C ω (real analytic), C [M ] (Denjoy-Carleman of Beurling or Roumieu type), C 0,1 (locally Lipschitz), or C k,α . The parameter domain E is either R or R n or an infinite dimensional convenient vector space. We completely describe the C -dependence on x of the eigenvalues and the eigenvectors of A(x). Thereby we extend previously known results for self-adjoint operators to normal operators, partly improve them, and show that they are best possible. For normal matrices A(x) we obtain partly stronger results.