Kato type operators and Weyl's theorem

A Banach space operator T satisfies Weyl's theorem if and only if T or T^∗ has SVEP at all complex numbers λ in the complement of the Weyl spectrum of T and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity. If T^∗ (respectively, T) has SVEP and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all λ∈isoσ(T)), then T satisfies a-Weyl's theorem (respectively, T^∗ satisfies a-Weyl's theorem).     

A note on generalized Weyl's theorem

We prove that if either T or T^∗ has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every f∈H(σ(T)). An application is given for algebraically paranormal operators.     

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.     

Property (w) and perturbations III

The property (w) is a variant of Weyl's theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T, whenever T is polaroid, or T has analytical core K(λ₀I−T)={0} for some λ₀∈C. The preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.     

Sequences of Hopf-Swan subgroups

Let l be an odd prime which satisfies Vandiver's conjecture, let n?1 be an integer, and let K=Q(ζn) where ζn is a primitive lnth root of unity. Let Cl denote the cyclic group of order l. For each j, j=1,…,ln−1, there exists an inclusion of Larson orders in KCl: Λj−1⊆Λj and a corresponding surjection of Hopf-Swan subgroups T(Λj−1)→T(Λj). For the cases n=1,2 we investigate the structure of various terms in the sequence of Hopf-Swan subgroups including the Swan subgroup T(Λ₀).     

Weyl type theorems for operators satisfying the single-valued extension property

Let T be a bounded linear operator acting on a Banach space X such that T or its adjoint T^∗ has the single-valued extension property. We prove that the spectral mapping theorem holds for the B-Weyl spectrum, and we show that generalized Browder's theorem holds for f(T) for every analytic function f defined on an open neighborhood U of σ(T). Moreover, we give necessary and sufficient conditions for such T to satisfy generalized Weyl's theorem. Some applications are also given.     

Generalized Browder's and Weyl's theorems for Banach space operators

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of σ(T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f∈H((T)), the space of functions analytic on an open neighborhood of σ(T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f∈H(σ(T)).     

The bitwisted Cartesian model for the free loop fibration

Using the notion of truncating twisting function from a simplicial set to a cubical set a special, bitwisted, Cartesian product of these sets is defined. For the universal truncating twisting function, the (co)chain complex of the corresponding bitwisted Cartesian product agrees with the standard Cartier (Hochschild) chain complex of the simplicial (co)chains. The modelling polytopes Fn are constructed. An explicit diagonal on Fn is defined and a multiplicative model for the free loop fibration ΩY→ΛY→Y is obtained. As an application we establish an algebra isomorphism H^∗(ΛY;Z)≈S(U)⊗Λ(s−1U) for the polynomial cohomology algebra H^∗(Y;Z)=S(U).     

On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

It is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σ_w(f(T)) = f(σ_w(T)), where σ(T) and σ_w(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T) + Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT + F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT + F.     

Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ=(λ₁,…,λk) with λ₁?n there is a Schubert variety X(λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T-equivariant cohomology of LGn and the T-equivariant fundamental class σ(λ) of X(λ). The main result of the present paper is an explicit formula for the restriction of the class σ(λ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LGn. As another consequence of the main result, we obtained a presentation of the ring .     

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.     

Higher rank numerical ranges and low rank perturbations of quantum channels

For a positive integer k, the rank-k numerical range Λk(A) of an operator A acting on a Hilbert space H of dimension at least k is the set of scalars λ such that PAP=λP for some rank k orthogonal projection P. In this paper, a close connection between low rank perturbation of an operator A and Λk(A) is established. In particular, for 1?r<k it is shown that Λk(A)⊆Λk−r(A+F) for any operator F with rank(F)?r. In quantum computing, this result implies that a quantum channel with a k-dimensional error correcting code under a perturbation of rank at most r will still have a (k−r)-dimensional error correcting code. Moreover, it is shown that if A is normal or if the dimension of A is finite, then Λk(A) can be obtained as the intersection of Λk−r(A+F) for a collection of rank r operators F. Examples are given to show that the result fails if A is a general operator. The closure and the interior of the convex set Λk(A) are completely determined. Analogous results are obtained for Λ_∞(A) defined as the set of scalars λ such that PAP=λP for an infinite rank orthogonal projection P. It is shown that Λ_∞(A) is the intersection of all Λk(A) for k=1,2,…. If A−μI is not compact for all μ∈C, then the closure and the interior of Λ_∞(A) coincide with those of the essential numerical range of A. The situation for the special case when A−μI is compact for some μ∈C is also studied.     

The Rotation Number Approach to the Periodic Fuc caron ik Spectrum

In this paper, we study the Fu?ik spectrum of the problem: (^*) ?+(λ₊+q₊(t))x₊+(λ₋+q₋(t))x₋=0 with the 2π-periodic boundary condition, where q_±(t) are 2π-periodic. After introducing a rotation number function ρ(λ₊, λ₋) for (^*), we prove using the Hamiltonian structure and the positive homogeneity of (^*) that for any positive integer n, the two boundary curves of the domain ρ⁻¹(n/2) in the (λ₊, λ₋)-plane are Fu?ik curves of (^*). The result obtained in this paper shows that such a spectrum problem is much like that of the higher dimensional Fu?ik spectrum with the Dirichlet condition. In particular, it remains open if the Fu?ik spectrum of (^*) is composed of only these curves.     

Singular continuous Floquet operator for systems with increasing gaps

Consider the Floquet operator of a time-independent quantum system, periodically perturbed by a rank one kick, acting on a separable Hilbert space: e^−iH₀Te^{−iκT|φ〉〈φ|}, where T and κ are the period and the coupling constant, respectively. Assume the spectrum of the self-adjoint operator H₀ is pure point, simple, bounded from below and the gaps between the eigenvalues (λ_n) grow like λ_n+1−λ_n∼Cn^d with d?2. Under some hypotheses on the arithmetical nature of the eigenvalues and the vector φ, cyclic for H₀, we prove the Floquet operator of the perturbed system has purely singular continuous spectrum.     

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).     

Generalized bi-circular projections

Recall that a projection P on a complex Banach space X is a generalized bi-circular projection if P+λ(I−P) is a (surjective) isometry for some λ such that |λ|=1 and λ≠1. It is easy to see that every hermitian projection is generalized bi-circular. A generalized bi-circular projection is said to be nontrivial if it is not hermitian. Botelho and Jamison showed that a projection P on C([0,1]) is a nontrivial generalized bi-circular projection if and only if P−(I−P) is a surjective isometry. In this article, we prove that if P is a projection such that P+λ(I−P) is a (surjective) isometry for some λ, then either P is hermitian or λ is an nth unit root of unity. We also show that for any nth unit root λ of unity, there are a complex Banach space X and a nontrivial generalized bi-circular projection P on X such that P+λ(I−P) is an isometry.     

The second largest eigenvalue of a tree

Denote by λ₂(T) the second largest eigenvalue of a tree T. An easy algorithm is given to decide whether λ₂(T)?λ for a given number λ, and a structure theorem for trees withλ₂(T)?λ is proved. Also, it is shown that a tree T with n vertices has $\text{λ}{}_2\text{(T)?lsqb}\text{(n?3)}\text{2}\text{rsqb}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$; this bound is best possible for odd n.     

Caractérisation spectrale des algèbres de Banach de dimension finie

Let f be an analytic function mapping a domain in $\text{C}$ into a complex Banach algebra. Using potential theory and a new result on almost continuity of the spectrum, which extends the theorem of Newburgh, we prove that either the set of λ such that the spectrum of f(λ) is finite is of outer capacity zero, or there exists an integer n such that the spectrum of f(λ) has at most n elements for every λ. From this we get extensions of a theorem given, in the complex case, by Kaplansky in 1954 and Hirschfeld and Johnson in 1972. More precisely we show that, if the spectrum is finite for every element of an open set of a real algebra or of the set of Hermitian elements of an algebra with an involution, then the quotient of this algebra by its radical is finite-dimensional.     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.