Relatively Bounded and Relatively Compact Perturbations for Limit Circle Hamiltonian Systems

We study Fefferman–Stein inequalities for the dyadic square function associated with an integrable, Hilbert-space-valued function on the interval [0, 1). The proof rests on a Bellman function method: the estimates are deduced from the existence of certain special functions enjoying appropriate majorization and concavity.     

Rank One Perturbations with Infinitesimal Coupling

We consider a positive self-adjoint operator A and formal rank one perturbations B = A + α(φ, ·)φ, where φ ₋₂(A) but φ ₋₁ (A), with _s(A) the usual scale of spaces. We show that B can be defined for such φ and what are essentially negative infinitesimal values of α. In a sense we will make precise, every rank one perturbation is one of three forms: (i) φ ₋₁(A), α ; (ii) φ ₋₁, α = ∞; or (iii) the new type we consider here.     

On the Point Spectrum of Self-Adjoint Operators That Appears under Singular Perturbations of Finite Rank

We discuss purely singular finite-rank perturbations of a self-adjoint operator A in a Hilbert space . The perturbed operators are defined by the Krein resolvent formula , Im z 0, where B _z are finite-rank operators such that dom B _z dom A = |0}. For an arbitrary system of orthonormal vectors satisfying the condition span | _i } dom A = |0} and an arbitrary collection of real numbers , we construct an operator that solves the eigenvalue problem . We prove the uniqueness of under the condition that rank B _z = n.     

Rank Properties in Finite Semigroups II: The Small Rank and the Large Rank

This paper is concerned with the application of two possible definitions of rank to certain well-known semigroups.AMS Subject Classification (2000), 20M10     

Rank One Perturbations in a Pontryagin Space with One Negative Square

Let N₁ denote the class of generalized Nevanlinna functions with one negative square and let N_1, 0 be the subclass of functions Q(z)∈N₁ with the additional properties lim_y→∞ Q(iy)/y=0 and lim sup_y→∞ y |Im Q(iy)|<∞. These classes form an analytic framework for studying (generalized) rank one perturbations A(τ)=A+τ[·, ω] ω in a Pontryagin space setting. Many functions appearing in quantum mechanical models of point interactions either belong to the subclass N_1, 0 or can be associated with the corresponding generalized Friedrichs extension. In this paper a spectral theoretical analysis of the perturbations A(τ) and the associated Friedrichs extension is carried out. Many results, such as the explicit characterizations for the critical eigenvalues of the perturbations A(τ), are based on a recent factorization result for generalized Nevanlinna functions.     

High Order Singular Rank One Perturbations of a Positive Operator

In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to with n ≥ 3, where is the scale of Hilbert spaces associated with L in      

The Spectrum and Trace Formula for Bounded Perturbations of Differential Operators

Spectrum properties and a method for deriving a regularized trace formula for perturbations of operators with discrete spectra in a separable Hilbert space are studied. A trace formula for a local perturbation of a two-dimensional harmonic oscillator in a strip is obtained based on this method.     

C0半群高阶微分算子的谱

本文研究了Ｃ０半群Ｔ（ｔ）的高阶微分算子Ｔ（ｎ）（ｔ）的谱给出了Ｔ＾（ｎ）（ｔ）的谱集的一种构造方法,讨论了Ｔ（ｎ）（ｔ）的谱点与Ｔ（ｔ）的无穷小生成元Ａ的谱点之间的关系。     

Measure Perturbations of Markovian Semigroups

The perturbation of the semigroup of a Borel right process by a class of signed measures on the state space of the process is studied. The perturbation is defined by a Feynman–Kac functional associated with the measure. Under appropriate conditions the perturbed semigroup is strongly continuous in L^p(m), 1 p< where m is a fixed excessive measure. Both existence and uniqueness of the associated Schrödinger type equation are investigated.     

A Non-Analytic Growth Bound for Laplace Transforms and Semigroups of Operators

Let f : \mathbbR₊ ? \mathbbC f : \mathbb{R}_{+} \longrightarrow \mathbb{C} be an exponentially bounded, measurable function. We introduce a growth bound z(f) \zeta(f) which measures the extent to which f f can be approximated by holomorphic functions. This growth bound is related to the location of the domain of holomorphy of the Laplace transform of f f far from the real axis. The denition extends to vector and operator-valued cases. For a C₀ C_{0} -semigroup T T of operators, z(T) \zeta(T) is closely related to the critical growth bound of T T .     

Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_, for A_, = A + (·), where A is a self-adjoint positive operator, being the A-scale). In the present note it is remarked that the operator A_, also appears directly as the Friedrichs extension of the symmetric operator :=A \{f (A)| f,=0\}. It is also shown that Krein's resolvents formula: (A_{_b,}-z)^-1 =(A-z)^-1+ (·, ) _z, with b=b-(1+z) (_z,_-1),_z= (A-z)^-1 defines a self-adjoint operator A_b, for each and b R¹. Moreover it is proven that for any sequence _n which goes to in there exists a sequence _n0 such that A_b, in the strong resolvent sense.     

Multiplicative Perturbations of Local C-Regularized Semigroups

In this paper, we investigate the multiplicative perturbations of local C-regularized semigroups on Banach spaces and establish some new multiplicative perturbation results which are generalizations of many existing theorems. Moreover, we give applications of the abstract results to three concrete problems including the mixed initial-boundary value problem for backwards heat equations with finite rank feedbacks.     

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.     

On One Class of Almost Bounded Perturbations of Smooth Restrictions of a Closed Operator

We investigate one class of perturbations of a closed densely-defined operator in a Hilbert space. These perturbations change the domain of definition of the operator. We prove that the perturbed operator S is closed and densely defined. We construct the adjoint operator S*.     

Finite Rank Perturbations of Toeplitz Operators

We study finite rank perturbations of the Brown-Halmos type results involving products of Toeplitz operators acting on the Bergman space.      

Growth of Rank 1 Valuation Semigroups

We consider semigroup S of a rank 1 valuation ? centered on a local ring R. We show that the Hilbert polynomial of R gives a bound on the growth of the valuation semigroup S. This allows us to give a very simple example of a well ordered subsemigroup of Q+, which is not a value semigroup of a local domain. We also consider the rates of growth which are possible for S. We show that quite exotic behavior can occur. In the final section, we consider the general question of characterizing rank 1 value semigroups.     

Markov积分半群的扰动和逼近

为了研究Markov积分半群的扰动和逼近,根据转移函数与Markov积分半群之间一一对应关系,以及转移函数的扰动和逼近,通过积分的方法,获得了Markov积分半群的广义Phillips扰动定理和Trotter-Kato逼近定理.     

Triviality of the Peripheral Point Spectrum of Positive Semigroups on Atomic Banach Lattices

Generalizing a result of Keicher [4] we show that generators of positive C ₀-semigroups on super-atomic Banach lattices have trivial peripheral point spectrum provided they satisfy a certain growth condition.