Virtual Eigenvalues of the High Order Schrödinger Operator II

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and We study the asymptotic behavior as of the non-bottom negative eigenvalues of H_γ, which are born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (virtual eigenvalues). To this end we use the Puiseux-Newton diagram for a power expansion of eigenvalues of some class of polynomial matrix functions. For the groups of virtual eigenvalues, having the same rate of decay, we obtain asymptotic estimates of Lieb-Thirring type.     

Riesz Idempotent and Algebraically M-hyponormal Operators

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let be the Riesz idempotent for λ₀, where D is a closed disk of center λ₀ which contains no other points of σ (T). In this note we show that E is self-adjoint and As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every (ii) a-Browder’s theorem holds for f(S) for every and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Additive Maps Preserving Local Spectrum

Let X be a complex Banach space, and let be the space of bounded operators on X. Given and x ∈ X, denote by σ_T (x) the local spectrum of T at x. We prove that if is an additive map such that

Uniformly summing sets of operators on spaces of vector–valued continuous functions

Let X, Y be Banach spaces. We say that a set is uniformly p–summing if the series is uniformly convergent for whenever (x_n) belongs to . We consider uniformly summing sets of operators defined on a -space and prove, in case X does not contain a copy of c₀, that is uniformly summing iff is, where T (φ x) = (T^#φ) x for all and x∈X. We also characterize the sets with the property that is uniformly summing viewed in . Received: 1 July 2005     

Upper Triangular Operator Matrices, SVEP and Browder, Weyl Theorems

A Banach space operator T ∈ B(χ) is polaroid if points λ ∈ iso σ(T) are poles of the resolvent of T. Let denote, respectively, the approximate point, the Weyl, the Weyl essential approximate, the upper semi–Fredholm and lower semi–Fredholm spectrum of T. For A, B and C ∈ B(χ), let M _C denote the operator matrix . If A is polaroid on , M ₀ satisfies Weyl’s theorem, and A and B satisfy either of the hypotheses (i) A has SVEP at points and B has SVEP at points , or, (ii) both A and A* have SVEP at points , or, (iii) A^* has SVEP at points and B ^* has SVEP at points , then . Here the hypothesis that λ ∈ π₀(M _C ) are poles of the resolvent of A can not be replaced by the hypothesis are poles of the resolvent of A. For an operator , let . We prove that if A* and B* have SVEP, A is polaroid on π ^a ₀(M _C) and B is polaroid on π ^a ₀(B), then .      

A boundary Morera theorem

LetD ⊂C ^N ,N ≥ 2 be a bounded open set withC ² boundary and letL be an open connected set of affine complex hyperplanes inC ^N containing a hyperplane that misses . LetE = ∪_Λ∈LΛ, Γ =E ∩bD. Suppose thatf ∈C(Γ) and assume that

Completeness of systems of entire functions in spaces of holomorphic functions

Letf be an entire function in . For a broad class of distribution densities of the set Λ, a scale of sufficient conditions for the completeness of the system of functions {f(λ×z):λ∈Λ},z∈E, where , in the spaceH(E) of holomorphic functions onE with respect to the topology of uniform convergence on compact subsets is given in terms of the mutual indicator of the functionf and the setE. These conditions are new already forn=1 even ifE is a disk. Translated fromMatematicheskie Zametki, Vol. 66, No. 4, pp. 603–616, October, 1999.     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

A Generalization of Orthogonal Factorizations in Graphs

Let G be a graph with vertex set V(G) and edge set E(G) and let g and f be two integer-valuated functions defined on V(G) such that g(x) ≤f(x) for all x∈V(G). Then a (g, f)-factor of G is a spanning subgraph H of G such that g(x) ≤d _H (x) ≤f(x) for all x∈V(G). A (g, f)-factorization of G is a partition of E(G) into edge-disjoint (g, f)-factors. Let = {F ₁, F ₂, ..., F _m } be a factorization of G and H be a subgraph of G with mr edges. If F _i , 1 ≤i≤m, has exactly r edges in common with H, then is said to be r-orthogonal to H. In this paper it is proved that every (mg + kr, mf−kr)-graph, where m, k and r are positive integers with k < m and g≥r, contains a subgraph R such that R has a (g, f)-factorization which is r-orthogonal to a given subgraph H with kr edges. This research is supported by the National Natural Science Foundation of China (19831080) and RSDP of China     

Weakly equicompact sets of operators defined on Banach spaces

Let X and Y be Banach spaces. We say that a set (the space of all weakly compact operators from X into Y) is weakly equicompact if, for every bounded sequence (x_n) in X, there exists a subsequence (x_k(n)) so that (Tx_k(n)) is weakly uniformly convergent for T ∈ M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if is collectively weakly compact, then M^* is weakly equicompact iff M^** x^**={T^** x^** : T ∈ M} is relatively compact in Y for every x^** ∈X^**; 2) weakly equicompact sets are precompact in for the topology of uniform convergence on the weakly null sequences in X. Received: 14 February 2005; revised: 1 June 2005     

Sesquitransitive and Localizing Operator Algebras

An algebra of operators on a Banach space X is said to be transitive if X has no nontrivial closed subspaces invariant under every member of the algebra. In this paper we investigate a number of conditions which guarantee that a transitive algebra of operators is “large” in various senses. Among these are the conditions of algebras being localizing or sesquitransitive. An algebra is localizing if there exists a closed ball B ∌ 0 such that for every sequence (x _n ) in B there exists a subsequence and a bounded sequence (A _k ) in the algebra such that converges to a non-zero vector. An algebra is sesquitransitive if for every non-zero z ∈ X there exists C > 0 such that for every x linearly independent of z, for every non-zero y ∈ X, and every there exists A in the algebra such that and ||Az|| ≤ C||z||. We give an algebraic version of this definition as well, and extend Jacobson’s density theorem to algebraically sesquitransitive rings. The second and the third authors were supported by NSERC.     

H ∞-calculus for the Stokes operator on unbounded domains

We consider the Stokes operator A on unbounded domains of uniform C ^1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces , 1 < q < ∞, where for q ≥ 2 and for q ∈ (1, 2). Moreover, it was shown that A has maximal L ^p -regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H ^∞-calculus in for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator. Received: 12 October 2007     

Some properties of essential spectra of a positive operator, II

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ_ew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ_ef(T), then for every a ≠ 0, where T₋₁ is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if , where , then an operator on D_i is band irreducible.      

Exponentiating a Bundle of Linear Operators

This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss the exponentiability of a multivalued map F: X⇉X expressible in the form F(x) = {Ax:A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painlevé–Kuratowski limit

On the numerical range of a Volterra operator in Lp

LetA be the linear operator inL _p (0, 1), 1<p<∞,p≠2, defined by ,x∈L _p (0, 1),s∈[0,1]. We show that the real values of numbers in the numerical range ofA have maximum , whereq=p/(p−1). This amounts to an inequality between integrals, for which we determine the case of equality.     

Virtual Eigenvalues of the High Order Schrödinger Operator I

We consider the Schr?dinger operator H_γ = ( − Δ)^l + γ V(x)· acting in the space $$L_2 (\mathbb{R}^d ),$$ where 2l ≥ d, V (x) ≥ 0, V (x) is continuous and is not identically zero, and $$\lim _{|{\mathbf{x}}| \to \infty } V({\mathbf{x}}) = 0.$$ We obtain an asymptotic expansion as $$\gamma \uparrow 0$$of the bottom negative eigenvalue of H_γ, which is born at the moment γ = 0 from the lower bound λ = 0 of the spectrum σ(H₀) of the unperturbed operator H₀ = ( − Δ)^l (a virtual eigenvalue). To this end we develop a supplement to the Birman-Schwinger theory on the process of the birth of eigenvalues in the gap of the spectrum of the unperturbed operator H₀. Furthermore, we extract a finite-rank portion Φ(λ) from the Birman- Schwinger operator $$X_V (\lambda ) = V^{\frac{1} {2}} R_\lambda (H_0 )V^{\frac{1}{2}} ,$$ which yields the leading terms for the desired asymptotic expansion.     

An Inertial Proximal Algorithm with Dry Friction: Finite Convergence Results

Let H be a Hilbert space and A, B: H ⇉ H two maximal monotone operators. In this paper, we investigate the properties of the following proximal type algorithm:

Extension of Floquet''''s theory to nonlinear quasiperiodic differential equations

In this paper, we consider the following autonomous system of differential equations: x = Ax f(x,θ), θ = ω, where θ∈Rm, ω = (ω1,…,ωm) ∈ Rm, x ∈ Rn, A ∈ Rn×n is a constant matrix and is hyperbolic, f is a C∞ function in both variables and 2π-periodic in each component of the vector e which satisfies f = O(||x||2) as x → 0. We study the normal form of this system and prove that under some proper conditions this system can be transformed to an autonomous system: x = Ax g(x), θ = ω. Additionally, the proof of this paper naturally implies the extension of Chen's theory in the quasi-periodic case.     

Riesz Idempotent and Weyl’s Theorem for w-hyponormal Operators

Let T be a w-hyponormal operator on a Hilbert space H, its Aluthge transform, λ an isolated point of the spectrum of T, and E_λ and the Riesz idempotents, with respect to λ, of T and respectively. It is shown that Consequently, E_λ is self-adjoint, and if λ ≠ 0. Moreover, it is shown that Weyl’s theorem holds for f(T), where f ∈ H(σ (T)).     

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.