3$\times$3阶上三角型算子矩阵的可能谱

Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.     

三阶上三角算子矩阵点谱,连续谱和剩余谱的扰动

设H_1,H_2,H_3为无穷维复可分Hilbert空间,记M_(D,E,F)F=(ADE0BF00C)∈B(H_1⊕H_2⊕H_3).给定A∈B(H1),B∈B(H_2),C∈B(H_3),结合分析方法与算子分块技巧给出了MD,E,F的点谱,连续谱和剩余谱随D,E,F扰动的完全描述.     

3×3上三角算子矩阵的Weyl型定理

设A∈B(H1),B∈B(H2),C∈B(H3)为给定的三个算子,用M(D,E,F)= 表示一个作用在H1(?)H2(?)H3上的3×3算子矩阵．本文首先给出存在算子D∈B(H2,H1),E∈B(H3,H1),F∈B(H3,H2),使得M(D,E,F)为上半Fredholm算子(下半Fredholm算子)的充要条件．同时研究了3×3算子矩阵 M(D,E,F)的Weyl定理,α-Weyl定理,Browder定理和α-Browder定理．     

Generalized Drazin spectrum of operator matrices

Let A ∈ B(X) and B ∈ B(Y), MC be an operator on Banach space X ⊕ Y given A C by MC =A generalized Drazin spectrum defined by σgD(T) = {λ∈ C : T-0 BλI is not generalized Drazin invertible} is considered in this paperIt is shown thatσgD(A) ∪σgD(B) = σgD(MC) ∪ WgD(A, B, C),where WgD(A, B, C) is a subset of σgD(A) ∩σgD(B) and a union of certain holes in σgD(MC).Furthermore, several sufficient conditions for σgD(A) ∪σgD(B) = σgD(MC) holds for every C ∈ B(Y, X) are given.     

Polar Sets and Relative Dimension for Generalized Brownian Sheet

Let{(t);t∈R_ ~N}be a d-dimensional N-parameter generalized Brownian sheet.Necessaryand sufficient conditions for a compact set E×F to be a polar set for(t,(t))are proved.It is also provedthat if 2N≤αd,then for any compact set ER_>~N,d-2/2 Dim E≤inf{dimF:F ∈ B(R~d),P{(E)∩F≠φ}>0}≤d-2/β DimE,and if 2N>αd,then for any compact set FR~d\{0},α/2(d-DimF)≤inf{dimE:E∈B(R_>~N),P{(E)∩F≠φ}>0}≤β/2(d-DimF),where B(R~d)and B(R_>~N)denote the Borel σ-algebra in R~d and in R_>~N respectively,dim and Dim are Hausdorffdimension and Packing dimension respectively.     

Semi-Fredholm Spectrum and Weyl＇s Theorem for Operator Matrices

When A ∈ B（H） and B ∈ B（K） are given, we denote by Mc an operator acting on the Hilbert space HΘ K of the form Me = （ A0 CB）. In this paper, first we give the necessary and sufficient condition for Mc to be an upper semi-Fredholm （lower semi-Fredholm, or Fredholm） operator for some C ∈B（K,H）. In addition, let σSF＋（A） = {λ ∈ C ： A-λI is not an upper semi-Fredholm operator} bc the upper semi-Fredholm spectrum of A ∈ B（H） and let σrsF- （A） = {λ∈ C ： A-λI is not a lower semi-Fredholm operator} be the lower semi Fredholm spectrum of A. We show that the passage from σSF±（A） U σSF±（B） to σSF±（Mc） is accomplished by removing certain open subsets of σSF-（A） ∩σSF＋ （B） from the former, that is, there is an equality σSF±（A） ∪σSF± （B） = σSF± （Mc） ∪＆ where L is the union of certain of the holes in σSF±（Mc） which ilappen to be subsets of σSF- （A） A σSF＋ （B）. Weyl＇s theorem and Browder＇s theorem are liable to fail for 2 × 2 operator matrices. In this paper, we also explore how Weyl＇s theorem, Browder＇s theorem, a-Weyl＇s theorem and a-Browder＇s theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.     

缺项算子矩阵的本性谱

Let =(A C X B)be a 2×2 operator matrix acting on the Hilbert space н( )κ.For given A ∈B (H),B ∈B(K)and C ∈B(K,H)the set Ux∈B(H,к)σe(Mx)is determined,where σe(T)denotes the essential spectrum.     

Property(ω) and Its Perturbations

A Hilbert space operator T is said to have property(ω1) if σa(T)\σaw(T) ? π00(T), where σa(T) and σaw(T) denote the approximate point spectrum and the Weyl essential approximate point spectrum of T respectively, and π00(T) = {λ∈ iso σ(T), 0 dim N(T- λI) ∞}. If σa(T)\σaw(T) = π00(T), we say T satisfies property(ω). In this note, we investigate the stability of the property(ω1) and the property(ω) under compact perturbations, and we characterize those operators for which the property(ω1) and the property(ω) are stable under compact perturbations.     

3×3上三角算子矩阵值域的闭性研究

令H_1,H_2,H_3是可分的复Hilbert空间,记M=(AEF0BD00C)为H_1⊕H_2⊕H_3上的3×3上三角算子矩阵.设A∈B(H_1),B∈B(H_2),C∈B(H_3)是给定的算子,利用对角元算子A,B,C的值域和零空间性质描述了算子矩阵M值域R(M)的闭性.     

Some new classes of Kadison-Singer lattices in Hilbert spaces

Let N be a maximal and discrete nest on a separable Hilbert space H,E the projection from H onto the subspace[C]spanned by a particular separating vector for N′and Q the projection from K=H⊕H onto the closed subspace{(,):∈H}.Let L be the closed lattice in the strong operator topology generated by the projections(E 00 0),{(E 00 0):E∈N}and Q.We show that L is a Kadison-Singer lattice with trivial commutant,i.e.,L′=CI.Furthermore,we similarly construct some Kadison-Singer lattices in the matrix algebras M2n(C)and M2n.1(C).     

The fractional metric dimension of permutation graphs

Let G =(V(G), E(G)) be a graph with vertex set V(G) and edge set E(G). For two distinct vertices x and y of a graph G, let RG{x, y} denote the set of vertices z such that the distance from x to z is not equa l to the distance from y to z in G. For a function g defined on V(G) and for U■V(G), let g(U) =∑s∈Ug(s). A real-valued function g : V(G) → [0, 1] is a resolving function of G if g(RG{x, y}) ≥ 1 for any two distinct vertices x, y ∈ V(G). The fractional metric dimension dimf(G)of a graph G is min{g(V(G)) : g is a resolving function of G}. Let G1 and G2 be disjoint copies of a graph G, and let σ : V(G1) → V(G2) be a bijection. Then, a permutation graph Gσ =(V, E) has the vertex set V = V(G1) ∪ V(G2) and the edge set E = E(G1) ∪ E(G2) ∪ {uv | v = σ(u)}. First,we determine dimf(T) for any tree T. We show that 1 dimf(Gσ) ≤1/2(|V(G)| + |S(G)|) for any connected graph G of order at least 3, where S(G) denotes the set of support vertices of G. We also show that, for any ε 0, there exists a permutation graph Gσ such that dimf(Gσ)- 1 ε. We give examples showing that neither is there a function h1 such that dimf(G) h1(dimf(Gσ)) for all pairs(G, σ), nor is there a function h2 such that h2(dimf(G)) dimf(Gσ) for all pairs(G, σ). Furthermore,we investigate dimf(Gσ) when G is a complete k-partite graph or a cycle.     

Browder spectra and essential spectra of operator matrices

Let Mc = （ A0CB ） be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that
σr（A） ∪ σr（ B）= σr （Mc） ∪ W ,
where W is the union of certain of the holes in σr（Mc） which happen to be subsets of σr（A） ∩ σr（B）, and σr（A）, σr（B）, σr（Mc） can be equal to the Browder or essential spectra of A, B, Mc, respectively. We also show that the above result isn＇t true for the Kato spectrum, left （right） essential spectrum and left （right） spectrum.     

Finite groups whose n-maximal subgroups are σ-subnormal

Let σ = {σ_i | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member ≠ 1 of H is a Hall σ_i-subgroup of G, for some i ∈ I, and H contains exactly one Hall σ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HA~x= A~xH for all A ∈ H and x ∈ G:σ-subnormal in G if there is a subgroup chain A = A_0≤A_1≤···≤ A_t = G such that either A_(i-1)■A_i or A_i/(A_(i-1))A_i is a finite σ_i-group for some σ_i ∈σ for all i = 1,..., t.If M_n M_(n-1) ··· M_1 M_0 = G, where Mi is a maximal subgroup of M_(i-1), i = 1, 2,..., n, then M_n is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal(σ-quasinormal,respectively) in G but, in the case n 1, some(n-1)-maximal subgroup is not σ-subnormal(not σ-quasinormal,respectively) in G, we write m_σ(G) = n(m_(σq)(G) = n, respectively).In this paper, we show that the parameters m_σ(G) and m_(σq)(G) make possible to bound the σ-nilpotent length l_σ(G)(see below the definitions of the terms employed), the rank r(G) and the number |π(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m_σ(G) = |π(G)|. Some known results are generalized.     

Symmetry of the point spectrum of infinite dimensional hamiltonian operators and its applications

This paper studies the symmetry, with respect to the real axis, of the point spectrum of the upper triangular infinite dimensional Hamiltonian operator H. Note that the point spectrum of H can be described as σp(H) = σp(A) ∪σp1(-A*). Using the characteristic of the set σp1(-A*), we divide the point spectrum σp(A) of A into three disjoint parts. Then, a necessary and sufficient condition is obtained under which σp1(-A*) and one part of σp(A) are symmetric with respect to the real axis each other. Based on this result, the symmetry of σp(H) is completely given. Moreover, the above result is applied to thin plates on elastic foundation, plane elasticity problems and harmonic equations.     

Brake Orbits of a Reversible Even Hamiltonian System Near an Equilibrium

In this paper, we consider the brake orbits of a reversible even Hamiltonian system near an equilibrium. Let the Hamiltonian system(H S) ■ = J H(x) satisfies H(0) = 0, H(0) = 0, reversible and even conditions H(Nx) = H(x) and H(-x) = H(x) for all x ∈ R²ⁿ. Suppose the quadratic form Q(x) =1/2 is non-degenerate. Fix τ₀ > 0 and assume that R²ⁿ= E ⊕ F decomposes into linear subspaces E and F which are invariant under the flow associated to the li...     

On a Proof of Roth''''s Theorem

We denote by M_(n,m)(F) the set of all n×m matrices over the field F and by M_n(F) the set of all n×n matrices over the field F. W. E. Roth has shown the following theorem in 1952, [1]. Theorem Let A∈M_n(F),B∈M_m(F) and C∈M_(n,m)(F), then the matrix equation AX-YB=C (1) has a solution X, Y∈M_(n,m)(F) if and only if the matrices     

3×3阶上三角算子矩阵的点谱、剩余谱和连续谱

记■为Hilbert空间■上的上三角算子矩阵.我们借助对角元A,B和C的谱性质给出了σ_*(M_(D,E,F))=σ_*(A)∪σ_*(B)∪σ_*(C)对任意D∈B(H_2,H_1),E∈B(H_3,H_1),F∈B(H_3,H_2)均成立的充要条件,其中σ_*代表某类特定的谱,如点谱、剩余谱和连续谱等.此外,给出了一些例证.     

ON DISTRIBUTIONAL n-CHAOS

Let(X, f) be a topological dynamical system, where X is a nonempty compact and metrizable space with the metric d and f : X → X is a continuous map. For any integer n ≥ 2, denote the product space by X(n)= X ×× X n times. We say a system(X, f) is generally distributionally n-chaotic if there exists a residual set D ? X(n)such that for any point x =(x1,, xn) ∈ D,lim infk→∞#({i : 0 ≤ i ≤ k- 1, min{d(fi(xj), fi(xl)) : 1 ≤ j = l ≤ n} δ0})k= 0for some real number δ0 0 and lim sup k→∞#({i : 0 ≤ i ≤ k- 1, max{d(fi(xj), fi(xl)) : 1 ≤ j = l ≤ n} δ})k= 1for any real number δ 0, where #() means the cardinality of a set. In this paper, we show that for each integer n ≥ 2, there exists a system(X, σ) which satisfies the following conditions:(1)(X, σ) is transitive;(2)(X, σ) is generally distributionally n-chaotic, but has no distributionally(n + 1)-tuples;(3) the topological entropy of(X, σ) is zero and it has an IT-tuple.     

Meromorphic Convex Functions with Negative Coefficients

Let σ_k(a) be the class of functions f(f)=1/z-sur from n=1 to ∞(|a_n|z~n), regular in the punctured disk E={z:0<|z|<1} and satisfying Re(1 zf"(z)/f'(z))<-a (0≤a<1) for z∈E. In this paper we obtain coefficient inequalities, distortion and closure Theorems for the class σ_k(a). Further we obtain the class preserving integral operator of the form     

THREE POINT BOUNDARY VALUE PROBLEMS FOR NONLINEAR FRACTIONAL DIFFERENTIAL EQUATIONS

In this paper, we study existence and uniqueness of solutions to nonlinear three point boundary value problems for fractional differential equation of the type c D δ 0+ u(t) = f (t, u(t), c D σ 0+ u(t)), t ∈ [0, T ], u(0) = αu(η), u(T ) = βu(η), where 1 < δ < 2, 0 < σ < 1, α, β∈ R, η∈ (0, T ), αη(1 -β) + (1-α)(T βη) = 0 and c D δ 0+ , c D σ 0+ are the Caputo fractional derivatives. We use Schauder fixed point theorem and contraction mapping principle to obtain existence and uniqueness results. Examples are also included to show the applicability of our results.