Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ(T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f(T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ(T) if and only if f(T^∗) satisfies Weyl’s theorem.     

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.     

Weyl's theorem for algebraically totally hereditarily normaloid operators

A Banach space operator T∈B(X) is said to be totally hereditarily normaloid, T∈THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q?1, T∈H(q), if the quasi-nilpotent part H₀(T−λ)=(T−λ)^−q(0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f(T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ(T), and T^∗ satisfies a-Weyl's theorem. If also T^∗ has the single valued extension property, then f(T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ(T) on which it is defined.     

Maps preserving peripheral spectrum of Jordan semi-triple products of operators

Let A and B be (not necessarily unital or closed) standard operator algebras on complex Banach spaces X and Y, respectively. For a bounded linear operator A on X, the peripheral spectrum σ_π(A) of A is the set σ_π(A)={z∈σ(A):|z|=max_ω∈σ(A)|ω|}, where σ(A) denotes the spectrum of A. Assume that Φ:A→B is a map the range of which contains all operators of rank at most two. It is shown that the map Φ satisfies the condition that σ_π(BAB)=σ_π(Φ(B)Φ(A)Φ(B)) for all A,B∈A if and only if there exists a scalar λ∈C with λ³=1 and either there exists an invertible operator T∈B(X,Y) such that Φ(A)=λTAT^-1 for every A∈A; or there exists an invertible operator T∈B(X^∗,Y) such that Φ(A)=λTA^∗T^-1 for every A∈A. If X=H and Y=K are complex Hilbert spaces, the maps preserving the peripheral spectrum of the Jordan skew semi-triple product BA^∗B are also characterized. Such maps are of the form A?UAU^∗ or A?UA^tU^∗, where U∈B(H,K) is a unitary operator, A^t denotes the transpose of A in an arbitrary but fixed orthonormal basis of H.     

Existence and regularity of extremal solutions for a mean-curvature equation

We study a class of mean curvature equations −Mu=H+λup where M denotes the mean curvature operator and for p?1. We show that there exists an extremal parameter λ^∗ such that this equation admits a minimal weak solutions for all λ∈[0,λ^∗], while no weak solutions exists for λ>λ^∗ (weak solutions will be defined as critical points of a suitable functional). In the radially symmetric case, we then show that minimal weak solutions are classical solutions for all λ∈[0,λ^∗] and that another branch of classical solutions exists in a neighborhood (λ_∗−η,λ^∗) of λ^∗.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

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.     

Generic uniqueness of equilibrium points

Let X be a nonempty, convex and compact subset of normed linear space E (respectively, let X be a nonempty, bounded, closed and convex subset of Banach space E and A be a nonempty, convex and compact subset of X) and f:X×X→R be a given function, the uniqueness of equilibrium point for equilibrium problem which is to find x^∗∈X (respectively, x^∗∈A) such that f(x^∗,y)≥0 for all y∈X (respectively, f(x^∗,y)≥0 for all y∈A) is studied with varying f (respectively, with both varying f and varying A). The results show that most of equilibrium problems (in the sense of Baire category) have unique equilibrium point.     

Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A_j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A₁) + f(A₂) + ? + f(A_m)? ? ? f(A₁ + A₂ + ? + A_m)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A₁) + f(A₂) + ? + f(A_m)? ? f(?A₁ + A₂ + ? + A_m?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.     

A concavity inequality for symmetric norms

We review some recent convexity results for Hermitian matrices and we add a new one to the list: Let A be semidefinite positive, let Z be expansive, Z^∗Z?I, and let f:[0,∞)→[0,∞) be a concave function. Then, for all symmetric norms

‖f(Z^∗AZ)‖?‖Z^∗f(A)Z‖.     

Annihilator-preserving maps, multipliers, and derivations

For a commutative subspace lattice L in a von Neumann algebra N and a bounded linear map f:N∩algL→B(H), we show that if Af(B)C=0 for all A,B,C∈N∩algL satisfying AB=BC=0, then f is a generalized derivation. For a unital C^∗-algebra A, a unital Banach A-bimodule M, and a bounded linear map f:A→M, we prove that if f(A)B=0 for all A,B∈A with AB=0, then f is a left multiplier; as a consequence, every bounded local derivation from a C^∗-algebra to a Banach A-bimodule is a derivation. We also show that every local derivation on a semisimple free semigroupoid algebra is a derivation and every local multiplier on a free semigroupoid algebra is a multiplier.     

Minimal ranks of some quaternion matrix expressions with applications

Suppose that p(X, Y) = A − BX − X^(∗)B^(∗) − CYC^(∗) and q(X, Y) = A − BX + X^(∗)B^(∗) − CYC^(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(X, Y) with respect to pair of matrices X and Y = Y^(∗), and the minimal rank formula of q(X, Y) with respect to pair of matrices X and Y = −Y^(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

Maps preserving the nilpotency of products of operators

Let B(X) be the algebra of all bounded linear operators on the Banach space X, and let N(X) be the set of nilpotent operators in B(X). Suppose ?:B(X)→B(X) is a surjective map such that A,B∈B(X) satisfy AB∈N(X) if and only if ?(A)?(B)∈N(X). If X is infinite dimensional, then there exists a map f:B(X)→C?{0} such that one of the following holds:

(a)

There is a bijective bounded linear or conjugate-linear operator S:X→X such that ? has the form A?S[f(A)A]S^-1.

(b)

The space X is reflexive, and there exists a bijective bounded linear or conjugate-linear operator S : X′ → X such that ? has the form A ? S[f(A)A′]S⁻¹.

If X has dimension n with 3 ? n < ∞, and B(X) is identified with the algebra M_n of n × n complex matrices, then there exist a map f:M_n→C?{0}, a field automorphism ξ:C→C, and an invertible S ∈ M_n such that ? has one of the following forms:

     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Q_ij = 1 for 0 ? j − i ? N − 1 and Q_ij = 0 otherwise. We prove that det(QQ^∗) is the minimum of det(RR^∗) over all complex matrices R with the same dimensions as Q satisfying ∣R_ij∣ ? 1 whenever Q_ij = 1 and R_ij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ^∗.     

Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

The index formula and the spectral shift function for relatively trace class perturbations

We compute the Fredholm index, index(DA), of the operator DA=(d/dt)+A on L²(R;H) associated with the operator path , where (Af)(t)=A(t)f(t) for a.e. t∈R, and appropriate f∈L²(R;H), via the spectral shift function ξ(⋅;A₊,A₋) associated with the pair (A₊,A₋) of asymptotic operators A_±=A(±∞) on the separable complex Hilbert space H in the case when A(t) is generally an unbounded (relatively trace class) perturbation of the unbounded self-adjoint operator A₋.We derive a formula (an extension of a formula due to Pushnitski) relating the spectral shift function ξ(⋅;A₊,A₋) for the pair (A₊,A₋), and the corresponding spectral shift function ξ(⋅;H₂,H₁) for the pair of operators in this relative trace class context,This formula is then used to identify the Fredholm index of DA with ξ(0;A₊,A₋). In addition, we prove that index(DA) coincides with the spectral flow of the family {A(t)}_t∈R and also relate it to the (Fredholm) perturbation determinant for the pair (A₊,A₋): with the choice of the branch of ln(detH(⋅)) on C₊ such thatWe also provide some applications in the context of supersymmetric quantum mechanics to zeta function and heat kernel regularized spectral asymmetries and the eta-invariant.