谱表示

<正> Stone M.对Hilbert空间中一个具有简单谱的自伴算子建立了谱表示定理,即有实轴上的有限Borel测度μ,使得同构于L~2(μ),同时变A为乘以自变量λ的算子.Jauch等([2])讨论了一列交换的自伴算子完全集谱表示定理,但要求一个关于测度绝对连续性的假定.此外,依据约化理论([3])可知,如果A是可分Hilbert空间的自伴     

向量值柯西问题的适定性

设A为复Banach空间X上的稠定闭算子.我们证明了若抽象柯西问题具有C-适定性,则{λ∈C:Reλ≥0）　ρ（A）且存在C>0,使得任给λ∈C,Reλ≥0,都有（λ-A）-1≤C/1+　λ.因而A必为X上某一有界解析半群的无穷小生成元.对于定义在有限区间[0,T]上的抽象柯西问题,我们亦得到了类似的结果.     

Rank theorems of operators between Banach spaces

Let E and F be Banach spaces, and B(E,F) all of bounded linear operators on E into F. Let T0∈B(E,F) with an outer inverse T#0∈B(F,E). Then a characteristic condition of S=(I+T#0(T-T0))-1T#0 with T∈B(E,F) and ‖T#0(T-T0)‖<1, being a generalized inverse of T, is presented, and hence, a rank theorem of operators on E into F is established (which generalizes the rank theorem of matrices to Banach spaces). Consequently, an improved finite rank theorem and a new rank theorem are deduced. These results will be very useful to nonlinear functional analysis.     

Über das wachstum von resolventen in der nähe einer isolierten singularität

Let L(λ) be a bundle of linear bounded operators between two Banach spaces. In this paper we study the behaviour of {L(λ)}^?1, if λ tends to λ_o and L(λ_o) is a Fredholm operator with index 0. We show that the growth of this resolvent can be described by the length of certain chains of generalized principal vectors; if L(λ) depends analytically on the parameter λ, we get a complete characterization for an isolated singularity of L, and also a Laurent expansion for the resolvent. Finally, we give applications to a broad class of bundles of bounded self-adjoint operators.     

劳勃生的特殊星像函数和特殊凸像函数

<正> 设函数w在单位圆 E_z:|z|<1上是正则的.假如f(z)在 E_z上是单叶的,那末 D_f=f(E_z)是 w 平面上单叶的区域.记这种单叶函数f(z)的全体为 S_p,S_1=S.若 D_f 以原点 w=0 为星形中心,就是说若 w_0∈D_f则缐段■整个地落在区域 D_f 中,称这种函数 f(z)是 E_z 中的星像函数,其特徵是在 E_z     

Heisenberg群上一类算子的有界性

设 F是λ阶正则的齐次分布,-Q≤λ<0。作者研究了Heisenberg群上的算子g*F在加幂权的Lebesgue空间和Herz型 Hardy空间上的有界性,其中 g是一恰当的函数。而且,本文还得到了由BMO(H~n)函数和线性算子T所生成的交换子[b,T]在Herz空间上的有界性。     

Two Parameter Asymptotic Spectra in the Uniformly Elliptic Case

In this article we study the abstract two parameter eigenvalue problem $$\begin{gathered} T_1 u_1 = \left( {\lambda _1 V_{11} + \lambda _2 V_{12} } \right)u_1 , \left\| {u_1 } \right\| = 1 \hfill \\ T_2 u_2 = \left( {\lambda _1 V_{21} + \lambda _2 V_{22} } \right)u_2 , \left\| {u_2 } \right\| = 1 \hfill \\ \end{gathered}$$ where, in the Hilbert spaces H_j, T_j is self-adjoint, bounded below and has compact resolvent, and V_jk are self-adjoint bounded operators, (?1)^j+kV_jk >> 0, j, k = 1, 2. An eigenvalue λ for this problem is a point in R² satisfying both equations. Under appropriate conditions, the eigenvalues λⁿ = (λ₁ ⁿ, λ₂ ⁿ) are countable and in R². We aim to describe the set of limit points of λⁿ/∥λⁿ∥, as ∥λⁿ∥ → ∞, in terms of the V_jk.     

集值映射的不动点指数与带间断非线性项的椭圆型方程的多重解

<正> 本文研究二阶半线性椭圆边值问题■的多重解(符号详见§3),其中φ(x,t)允许对t是不连续的.一些自由边界问题可以化归这类问题.为了统一处理φ(x,t)对t连续与不连续两种情形,我们采用集值映射的观点.为此推广了经典的算子与Hammerstein算子到集值映射,并发展了集值映射的Leray-Schauder度理论;与已有的集值映射理论不同,现在处理的是映射串(定     

The surjectivity radius,packing numbers and boundedness below of linear operators

Let T be a bounded linear operator on a Banach space. Denote by r(T) the supremum of all ε≥O such that the operator T - λI is surjective for |λ|<ε. Similarly let b(T) be the supremum of all ε≥O such that the operator T - λI is bounded below for |λ|<ε. We prove formulas for r(T) and b(T) which are analogous to the classical spectral radius formula. We also obtain related results with the packing numbers of T, an application concerning topological divisors of zero in Banach algebras, and finally we prove the lower semi-continuity of the functions r(·) and b(·).     

Multiplicative Decompositions of Holomorphic Fredholm Functions and ψ*-Algebras

In this article we construct multiplicative decompositions of holomorphic Fredholm operator valued functions on Stein manifolds with values in various algebras of differential and pseudo differential operators which are submultiplicative ψ* - algebras, a concept introduced by the first author. For Fredholm functions T(z) satisfying an obvious topological condition we. Prove (0.1) T(z) = A(z)(I + S(z)), where A(z) is holomorphic and invertible and S(z) is holomorphic with values in an “arbitrarily small” operator ideal. This is a stronger condition on S(z) than in the authors' additive decomposition theorem for meromorphic inverses of holomorphic Fredholm functions [12], where the smallness of S(z) depends on the number of complex variables. The Multiplicative Decomposition theorem (0.1) sharpens the authors' Regularization theorem [11]; in case of the Band algebra L(X) of all bounded linear operators on a Band space, (0.1) has been proved by J. Letterer [20] for one complex variable and by M. 0. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov [26] for the Banach ideal of compact operators.