Numerical ranges of nilpotent operators

For any operator A on a Hilbert space, let W(A), w(A) and w₀(A) denote its numerical range, numerical radius and the distance from the origin to the boundary of its numerical range, respectively. We prove that if Aⁿ=0, then w(A)?(n-1)w₀(A), and, moreover, if A attains its numerical radius, then the following are equivalent: (1) w(A)=(n-1)w₀(A), (2) A is unitarily equivalent to an operator of the form aA_n⊕A^′, where a is a scalar satisfying |a|=2w₀(A), A_n is the n-by-n matrix

     

On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.     

Singular value estimates of oblique projections

Let W and M be two finite dimensional subspaces of a Hilbert space H such that H=W⊕M^⊥, and let P_W‖M^⊥ denote the oblique projection with range W and nullspace M^⊥. In this article we get the following formula for the singular values of P_W‖M^⊥

     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

On weakly unitarily invariant norm and the λ-Aluthge transformation for invertible operator

Let T∈B(H) be an invertible operator with polar decomposition T = UP and B∈B(H) commute with T. In this paper we prove that ∣∣∣P^λBUP^1−λ∣∣∣ ? ∣∣∣BT∣∣∣, where ∣∣∣ · ∣∣∣ is a weakly unitarily invariant norm on B(H) and 0 ? λ ? 1. As the consequence of this result, we have ∣∣∣f(P^λUP^1−λ)∣∣∣ ? ∣∣∣f(T)∣∣∣ for any polynomial f.     

Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

We study the spectral shift function s(λ,h) and the resonances of the operator P(h)=-Δ+V(x)+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s(λ,h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s(λ,h). We establish an upper bound O(h^-n+1) for the number of the resonances of P(h) lying in a disk of radius h.     

Reflexivity defect of spaces of linear operators

For a finite-dimensional linear subspace S⊆L(V,W) and a positive integer k, the k-reflexivity defect of S is defined by rd_k(S)=dim(Ref_k(S)/S), where Ref_k(S) is the k-reflexive closure of S. We study this quantity for two-dimensional spaces of operators and for single generated algebras and their commutants.     

Isometries of a generalized numerical radius

For 0<q<1, the q-numerical range is defined on the algebra M_n of all n×n complex matrices by

W_q(A)={x^∗Ay:x,y∈Cⁿ,∥x∥=∥y∥=1,〈y,x〉=q}.     

Maps completely preserving spectral functions

Let X,Y be infinite dimensional complex Banach spaces and A,B be standard operator algebras on X and Y, respectively. In this paper, we show that surjective maps completely preserving certain spectral function Δ(·) from A to B are isomorphisms, where Δ(·) stands for any one of 13 spectral functions σ(·), σ_l(·), σ_r(·), σ_l(·)∩σ_r(·), ∂σ(·), ησ(·), σ_p(·), σ_c(·), σ_p(·)∩σ_c(·), σ_p(·)∪σ_c(·), σ_ap(·), σ_s(·), and σ_ap(·)∩σ_s(·).     

A counterexample to a conjecture of Matsaev

We present an effective algorithm for estimating the norm of an operator mapping a low-dimensional ?^p space to a Banach space with an easily computable norm. We use that algorithm to show that Matsaev’s proposed extension of the inequality of John von Neumann is false in case p=4. Matsaev conjectured that for every contraction T on L^p (1<p<∞) one has for any polynomial P

‖P(T)‖_L^p→L^p?‖P(S)‖_{?^p(Z⁺)→?^p(Z⁺)}     

An algorithm for computing Jordan chains and inverting analytic matrix functions

We present an efficient algorithm for obtaining a canonical system of Jordan chains for an n × n regular analytic matrix function A(λ) that is singular at the origin. For any analytic vector function b(λ), we show that each term in the Laurent expansion of A(λ)⁻¹b(λ) may be obtained from the previous terms by solving an (n + d) × (n+d) linear system, where d is the order of the zero of det A(λ) at λ = 0. The matrix representing this linear system contains A(0) as a principal submatrix, which can be useful if A(0) is sparse. The last several iterations can be eliminated if left Jordan chains are computed in addition to right Jordan chains. The performance of the algorithm in floating point and exact (rational) arithmetic is reported for several test cases. The method is shown to be forward stable in floating point arithmetic.     

Multiplicative mappings at some points on matrix algebras

Let M_n be the algebra of all n×n matrices, and let φ:M_n→M_n be a linear mapping. We say that φ is a multiplicative mapping at G if φ(ST)=φ(S)φ(T) for any S,T∈M_n with ST=G. Fix G∈M_n, we say that G is an all-multiplicative point if every multiplicative linear bijection φ at G with φ(I_n)=I_n is a multiplicative mapping in M_n, where I_n is the unit matrix in M_n. We mainly show in this paper the following two results: (1) If G∈M_n with detG=0, then G is an all-multiplicative point in M_n; (2) If φ is an multiplicative mapping at I_n, then there exists an invertible matrix P∈M_n such that either φ(S)=PSP^-1 for any S∈M_n or φ(T)=PT^trP^-1 for any T∈M_n.     

The semiclassical resolvent and the propagator for non-trapping scattering metrics

Consider a compact manifold with boundary M with a scattering metric g or, equivalently, an asymptotically conic manifold (M^○,g). (Euclidean Rn, with a compactly supported metric perturbation, is an example of such a space.) Let Δ be the positive Laplacian on (M,g), and V a smooth potential on M which decays to second order at infinity. In this paper we construct the kernel of the operator ⁻¹(h²Δ+V−²(λ₀±i0)), at a non-trapping energy λ₀>0, uniformly for h∈(0,h₀), h₀>0 small, within a class of Legendre distributions on manifolds with codimension three corners. Using this we construct the kernel of the propagator, e−it(Δ/2+V), t∈(0,t₀) as a quadratic Legendre distribution. We also determine the global semiclassical structure of the spectral projector, Poisson operator and scattering matrix.     

First order spectral perturbation theory of square singular matrix polynomials

We develop first order eigenvalue expansions of one-parametric perturbations of square singular matrix polynomials. Although the eigenvalues of a singular matrix polynomial P(λ) are not continuous functions of the entries of the coefficients of the polynomial, we show that for most perturbations they are indeed continuous. Given an eigenvalue λ₀ of P(λ) we prove that, for generic perturbations M(λ) of degree at most the degree of P(λ), the eigenvalues of P(λ)+?M(λ) admit covergent series expansions near λ₀ and we describe the first order term of these expansions in terms of M(λ₀) and certain particular bases of the left and right null spaces of P(λ₀). In the important case of λ₀ being a semisimple eigenvalue of P(λ) any bases of the left and right null spaces of P(λ₀) can be used, and the first order term of the eigenvalue expansions takes a simple form. In this situation we also obtain the limit vector of the associated eigenvector expansions.     

Remarks on ideal boundedness, convergence and variation of sequences

We answer the questions asked by Faisant et al. (2005) [2]. The first main result states that for every admissible ideal I⊂P(N) the quotient space l^∞(I)/c₀(I) is complete. The second main result states that consistently there is an admissible ideal I⊂P(N) such that the sets W(I), of all real sequences with finite I-variation, and c^?(I), of all restrictively I-convergent sequences, are equal.     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

ENSS' Theory in Long Range Scattering: Second Order Hyperbolic and Parabolic Operators

We prove asymptotic completeness using ENSS' method for h₀(P) + W_S(Q) + W_L(Q) where h₀: Rⁿ → R is a polynomial of degree 2 with lim lim_|ξ|→∞h₀(ξ)| + |Δh₀(ξ)| = ∞, W_S a short range potential and W_L a smooth long range potential.     

Points d’évaluation pour les opérateurs cycliques ayant la propriété de Bishop (β)

Let T be a linear bounded cyclic operator in a separable complex Hilbert space H. Let B(T) and B_a(T) denote, respectively, the set of bounded point evaluation and the set of analytic point evaluation of T. We show that if T has the Bishop property (β), then B_a(T)=B(T)?σ_ap(T), where σ_ap(T) is the approximate spectrum of T. In the particular case when T is an operator of multiplication by z in a Hardy space this was proved by Trent (Pacific J. Math. 80 (1979) 279). On the other hand, using the generalized and the local spectral theory we obtain sufficient conditions on B_a(T) under which the spectrum of T and the local spectrum of T at any y≠0 in H coincide. At the end results involving the spectral picture of quasi-similar cyclic operators are given.     

The Dirichlet-to-Neumann operator on rough domains

We consider a bounded connected open set Ω⊂Rd whose boundary Γ has a finite (d−1)-dimensional Hausdorff measure. Then we define the Dirichlet-to-Neumann operator D₀ on L₂(Γ) by form methods. The operator −D₀ is self-adjoint and generates a contractive C₀-semigroup S=(St)_t>0 on L₂(Γ). We show that the asymptotic behaviour of St as t→∞ is related to properties of the trace of functions in H¹(Ω) which Ω may or may not have.