A canonical form for self-adjoint pencils in Hilbert space

2007

S Subadditive and Superadditive Inequalities T. Ando Faculty of Economics Hokusei Gakuen University Sapporo 004-8631 Japan [email protected] Key identifying words: subadditive and superadditve inequalities, operator-monotone function. It is known that if f(t) is a non-negative, operator-monotone (that is, matrix-monotone of all orders) function on [0;1) then the inequality jjjf(A+B) f(A)jjj jjjf(B)jjj holds for every pair of positive semi-de nite matrices A;B and every unitarily invariant norm jjj jjj. In this talk we discuss inequalities of the form jjjf(A+B)jjj jjjf(A) + f(B)jjj: Diagonals of Matrices Rajendra Bhatia Indian Statistical Institute New Delhi 110 016 India [email protected] Key identifying words: diagonal, norm, Fourier series, inequalities The diagonal of a matrix can be expressed as an average over unitary conjugates of the matrix.We will display such expressions for the main diagonal and other diagonals parallel to it.This leads to some interesting sharp bounds...

Proceedings of the American Mathematical Society, 1974

Various theorems on positive matrices are shown to be corollaries of one general theorem, the proof of which bears on Legendre functions, as used in Rockafellar's Convex analysis. 1. Introduction: The main theorem. Let X be a finite set, ([¿(x))xeX strictly positive numbers, and H a fixed linear subspace of Rx. We shall prove the following : Theorem 1. There exists a unique {nonlinear) map from Rx to H, denoted ft->hf, such that 2 texp/(x)-exp hf(x)]g(x)/¿(x) = 0 xeX for all g in H. 2. A first application: Matrices with prescribed marginals. Given an «-sequence /• = (ri)"=1 and an /w-sequence s=(sj),J!=1 of nonnegative numbers such that 22-i ^i=Yj=\ •*/> denote by <J?(r, s) the set of (n, m) matrices (aiS) with a^O such that ri=^JL1aij and Sj-^X-iO^ for all 1 = 1,2, •••,« and j=l,2, • • • , m. Also, let E={1,2, • ■ ■ ,n} and F={1, 2, ■ • •, m}. We define the linear map c from RE®RF to RExF by: c[(b¡)ieE> (t>'j)jeF] = (bi + b'i)(i.i)eExF-If A' is a subset of Ex F, tt denotes the canonical map from RExF to Rx, i.e. 7T[{ai})tí,i)eExF\-(aii)(i.i)eX-We say also that X is an (r, s) pattern if there exists (ai}) in ^#(r, s) such thatX={(ij);a«>0)-Now the first corollary to the Theorem 1 is : Corollary 1. Let two sequences r=(r$=1 ands=(s,)™=l of nonnegative numbers with 2?=i r¿=2í=i sn M an (n,m) matrix with /¿¿3-=0 and

Integral Equations and Operator Theory, 1978

Note di Matematica

The classical Nagy-Foiaş-Langer decomposition of an ordinary contraction is generalized in the context of the operators T on a complex Hilbert space H which, relative to a positive operator A on H, satisfy the inequality T*AT ≤ A. As a consequence, a version of the classical von Neumann-Wold decomposition for isometries is derived in this context. Also one shows that, if T*AT = A and AT = A1/2TA1/2, then the decomposition of H in normal part and pure part relative to A1/2T is just a von Neumann-Wold type decomposition for A1/2T, which can be completely described. As applications, some facts on the quasi-isometries recently studied in [4], [5], are obtained.

2021

The Kato's decomposition [11, Theorem 4] is generalized to semi-B-Fredholm operators. 1 Introduction and preliminaries Let T ∈ L(X); where L(X) is the Banach algebra of bounded linear operators acting on an infinite dimensional complex Banach space X. We denote by T * the dual of T, by α(T) the dimension of the kernel N (T) and by β(T) the codimension of the range R(T). A subspace M of X is Tinvariant if T (M) ⊂ M and in this case T M means the restriction of T on M. We say that T is completely reduced by a pair (M, N) ((M, N) ∈ Red(T) for brevity) if M and N are closed T-invariant subspaces of X and X = M ⊕ N ; here M ⊕ N means that M ∩ N = {0}. Let n ∈ N, we denote by T [n] := T R(T n) (in particular, T [0] = T). It is clear that (α(T [n])) n and (β(T [n])) n are decreasing sequences. We call m T := inf{n ∈ N : inf{α(T [n]), β(T [n])} < ∞} (with inf∅ = ∞) the essential degree of T. Following [7] we say that T has finite essential ascent (resp., descent) if a e (T) := inf{n ∈ N : α(T [n]) < ∞} < ∞ (resp., d e (T) := inf{n ∈ N : β(T [n]) < ∞} < ∞). Note that m T = inf{a e (T), d e (T)}. Moreover, if T has finite essential ascent and essential descent then m T = a e (T) = d e (T). Operators with finite essential ascent or descent seem to have been first studied in [4], these operators played an important role in many papers, see for example [1, 3, 5, 7]. It is easily seen that the definition given in [1, 3] of an upper semi-B-Fredholm (resp., lower semi-B-Fredholm) operator T is equivalent to the following definition given in [7]: a e (T) < ∞ and R(T ae(T)+1) is closed (resp., d e (T) < ∞ and R(T de(T)) is closed). T is called semi-B-Fredholm (resp., B-Fredholm) if it is an upper or (resp., and) a lower semi-B-Fredholm. According to Corollary 2.3 below, if T is a semi-B-Fredholm then R(T mT) is closed and m T = min{n ∈ N : R(T n) is closed and inf{α(T [n]), β(T [n])} < ∞}, and in this case the index of T is defined by ind(T) := α(T [mT ]) − β(T [mT ]). If T is an upper semi-B-Fredholm (resp., lower semi-B-Fredholm, semi-B-Fredholm, B-Fredholm) with essential degree m T = 0, then T is said to be an upper semi-Fredholm (resp., lower semi-Fredholm, semi-Fredholm, Fredholm) operator. The degree of stable iteration d := dis(T) of T is defined as d = inf ∆(T); where

Advances in Mathematics, 1996

This article concems the spectral theory of many classes of oper-ators defined by means of some inequalities. Particular emphasis is given to the Fredholm theory and local spectral theory of these classes of operators.

Linear and Multilinear Algebra, 2018

In this paper, we show that if T ∈ L(X) is a quasi-Fredholm operator of degree d such that N(T d) + R(T) is complemented, then T has a decomposition like the Kato type operators. Using this decomposition allows us get a result about the stability of this class of operators under perturbations by nilpotent operators. Also we give a new decomposition for the class of semi B-Weyl operators, and through this property we shown that T is a semi B-Weyl operator if and only if T = S+K where S is a semi B-Browder operator and K is finite-dimensional.

The main task of the paper is to demonstrate that Corollary 6 in [R.E. Hartwig, K. Spindelböck, Matrices for which A * and A † commute, Linear and Multilinear Algebra 14 (1984) 241-256] provides a powerful tool to investigate square matrices with complex entries. This aim is achieved, on the one hand, by obtaining several original results involving square matrices, and, on the other hand, by reestablishing some of the facts already known in the literature, often in extended and/or generalized forms. The particular attention is paid to the usefulness of the aforementioned corollary to characterize various classes of matrices and to explore matrix partial orderings.

Studia Scientiarum Mathematicarum Hungarica, 2010

Following a 1939 article of Favard we consider the composition of classical Bernstein operators and piecewise linear interpolation at mutually distinct knots in [0, 1], not necessarily equidistant. We prove direct theorems in terms of the classical and the Ditzian-Totik modulus of second order. 2010 Mathematics Subject Classification: 41A10, 41A15, 41A17, 41A25, 41A36. Key words and phrases: Favard-Bernstein operator, Bernstein polynomials for arbitrary points of interpolation, piecewise linear interpolation, Bernstein operator, second order modulus of smoothness, Ditzian-Totik modulus, positive linear operator, degree of approximation.

Linear Algebra and its Applications, 2020

Let U be a semiunitary space; i.e., a complex vector space with scalar product given by a positive semidefinite Hermitian form x¨,¨y. If a linear operator A : U Ñ U is bounded (i.e., }Au} ď c}u} for some c P R and all u P U), then the subspace U 0 :" tu P U | xu, uy " 0u is invariant, and so A defines the linear operators A 0 : U 0 Ñ U 0 and A 1 : U{U 0 Ñ U{U 0. Let A be an indecomposable bounded operator on U such that 0 ‰ U 0 ‰ U. Let λ be an eigenvalue of A 0. We prove that the algebraic multiplicity of λ in A 1 is not less than the geometric multiplicity of λ in A 0 , and the geometric multiplicity of λ in A 1 is not less than the number of Jordan blocks J t pλq of each fixed size tˆt in the Jordan canonical form of A 0. We give canonical forms of selfadjoint and isometric operators on U, and of Hermitian forms on U. For an arbitrary system of semiunitary spaces and linear mappings on/between them, we give an algorithm that reduces their matrices to canonical form. Its special cases are Belitskii's and Littlewood's algorithms for systems of linear operators on vector spaces and unitary spaces, respectively.

Numerical Linear Algebra with Applications, 2007

A matrix with positive row sums and all its off-diagonal elements bounded above by their corresponding row averages is called a B-matrix by J. M. Peña in References (SIAM J. Matrix Anal. Appl. 2001; 22:1027-1037) and (Numer. Math. 2003 95:337-345). In this paper, it is generalized to more extended matrices-M B-matrices, which is proved to be a subclass of the class of P-matrices. Subsequently, we establish relationships between defined and some already known subclasses of P-matrices (see, References SIAM 225:117-123). As an application, some subclasses of P-matrices are used to localize the real eigenvalues of a real matrix.

Semigroup Forum, 2009

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if A generates a polynomially bounded n-times integrated group whose spectrum set σ(A) = {iλ k ; k ∈ Z * } is discrete and satisfies 1 |λ k | ℓ δ n k < ∞ (n and ℓ nonnegative integers), then there exists projectors (P k) k∈Z * such that P k x = x (x ∈ D(A n+ℓ)), where δ k = min |λ k+1 −λ k | 2 , |λ k−1 −λ k | 2 .

2005

In this paper we are interested in spectral decomposition of an unbounded operator with discrete spectrum. We show that if A generates a polynomially bounded n-times integrated group whose spectrum set σ(A) = {iλ k ; k ∈ Z * } is discrete and satisfies 1 |λ k | ℓ δ n k < ∞ (n and ℓ nonnegative integers), then there exists projectors (P k) k∈Z * such that P k x = x (x ∈ D(A n+ℓ)), where δ k = min |λ k+1 −λ k | 2 , |λ k−1 −λ k | 2 .

Bulletin (New Series) of the American Mathematical …, 1986

A linear transformation T acting on a finite-dimensional complex vector space SC can always be decomposed as T = D + N, where (i) D is diagonalizable and N is nilpotent; and (ii) DN = ND\ moreover, such a decomposition is unique with respect to the conditions (i) and (ii), and both D and N are indeed polynomials in T. When % is an infinite-dimensional Banach space, such a representation for a bounded operator T is no longer true, but an important class of transformations introduced and studied by N. Dunford [3] in the 1950s possesses a similar property. By definition, a spectral operator T acting on 3C is one for which there exists a spectral measure E (i.e., a homomorphism from the Boolean algebra of Borel subsets of the complex plane C into the Boolean algebra of projection operators on & such that E is bounded and E(C) = /) satisfying the following two properties: (1) TE(B) = E(B)T\ and (2) a(T\ E(B)sr)