Singular Perturbations of Self-Adjoint Operators

Singular relatively compact perturbations of self-adjoint operators are studied. The results obtained are applied to the Schrödinger operator with a singular potential.     

Partial Multiplication of Operators in Rigged Hilbert Spaces

The problem of the multiplication of operators acting in rigged Hilbert spaces is considered. This is done, as usual, by constructing certain intermediate spaces through which the product can be factorized. In the special case where the starting space is the set of C-vectors of a self-adjoint operator A, a general procedure for constructing a special family of interspaces is given. Their definition closely reminds that of the Bessel potential spaces, to which they reduce when the starting space is the Schwartz space Some applications are considered.     

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.     

Lévy--Laplace Operators in Functional Rigged Hilbert Spaces

Mathematical Notes -     

Pseudodifferential Operators on Spaces of Distributions Associated with Non-negative Self-Adjoint Operators

We consider Hörmander type symbols on a family of spaces associated with non-negative self-adjoint operators, and we prove boundedness of the corresponding pseudodifferential operators on both classical and non-classical Besov and Triebel–Lizorkin spaces. Consequently, this also covers the case of Sobolev spaces. As an application, we obtain boundedness of spectral multipliers on the mentioned spaces.     

Perturbations of Invertible Operators and Stability of g-Frames in Hilbert Spaces

In this paper, we study the perturbations of invertible operators and stability of g-frames in Hilbert spaces. In particular, we obtain some conditions under which the perturbations of an invertible operator are still an invertible operator, the perturbations of a right invertible operator or a surjective operator are still a right invertible operator or surjective operator. Then we apply the perturbations of invertible operators to study the stability of g-frames which is close related with the invertibility (or right invertibility) property of operators.     

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.     

Similarity of Some Singular Operators to Self-Adjoint Ones

The singular differential operator is studied. It is proved that if the second moment of p is finite and L has no nonreal eigenvalues, then L is similar to a self-adjoint operator. The proof is based on an integral resolvent criterion of similarity applied to a wide class of functions p(x). Bibliography: 20 titles.     

空间H中G-框架的扰动

本文运用算子理论方法，讨论了Hilbert 空间$H$中$g$-框架和$g$-框架算子的性质; 并且研究了$g$-框架的扰动，给出了一些有意义的结果.     

奇型Sturm-Liouville微分算子的限界自伴扩张

本文给出了奇型Sturm—Liouville微分算子限界自伴扩张的充要条件,从而得 到按边值条件分类的所有限界自伴边值条件,并直接回答了奇型Sturm—Liouville问题 的最小特征值不等式中相等的边值条件.     

Atomic and Molecular Decomposition of Homogeneous Spaces of Distributions Associated to Non-negative Self-Adjoint Operators

We deal with homogeneous Besov and Triebel–Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel–Lizorkin spaces. Spectral multipliers for these spaces are established as well.

     

Compact and Finite Rank Perturbations of Closed Linear Operators and Relations in Hilbert Spaces

For closed linear operators or relations A and B acting between Hilbert spaces and the concepts of compact and finite rank perturbations are defined with the help of the orthogonal projections P _A and P _B in onto the graphs of A and B. Various equivalent characterizations for such perturbations are proved and it is shown that these notions are a natural generalization of the usual concepts of compact and finite rank perturbations. Sadly, our colleague and friend Peter Jonas passed away on July, 18th 2007.     

A Class of Singular Self-Adjoint Ordinary Differential Operators with Maximal Spectrum

We present a characterization of the almost everywhere convergence of the partial Fourier series of functions in L^p(T), 1 < p < ∞, in terms of a discrete weak-type inequality.     

具有内部奇异点的实系数微分算子的自共轭域

研究了一类带有内部奇异点的实系数微分算子自共轭域的描述问题.通过构造相应的直和空间,应用直和空间的相关理论及对相应最大算子域进行分解,在直和空间上生成的相应最小算子具有实正则型域的情形下,利用微分方程的实参数解给出此类算子的自共轭域的完全解析描述,并且确定其边界条件的矩阵仅由微分方程的解在正则点的初始值决定.     

On Symmetrizable Operators on Hilbert Spaces

This paper, motivated by transport theory, deals with spectral properties of operators G on a complex Hilbert space H such that SG is self-adjoint where S is a nonnegative operator: We give several lower bounds of the spectral radius of G and determine the latter in some cases. We derive the whole spectrum for power compact G by means of Lagrange multiplier theory. We find out spectral connections between G and SG. We give a (spectral) stability estimate for symmetrizable operators in terms of the spectral radius of the perturbation.     

Singular Integral Operators on Besov Spaces

Summary. We introduce generalized BESOV spaces in terms of mean oscillation and weight functions, following a recent work of Dorronsoro, and study the continuity of singular integral operators on them. Relations between these spaces and the BESOV spaces in terms of modulus of continuity are also studied. An application to pseudo-differential operators is given.     

Symmetric Operators and Reproducing Kernel Hilbert Spaces

We establish the following sufficient operator-theoretic condition for a subspace S ì L² (\mathbbR, dn){S \subset L^2 (\mathbb{R}, d\nu)} to be a reproducing kernel Hilbert space with the Kramer sampling property. If the compression of the unitary group U(t) := e ^itM generated by the self-adjoint operator M, of multiplication by the independent variable, to S is a semigroup for t ≥ 0, if M has a densely defined, symmetric, simple and regular restriction to S, with deficiency indices (1, 1), and if ν belongs to a suitable large class of Borel measures, then S must be a reproducing kernel Hilbert space with the Kramer sampling property. Furthermore, there is an isometry which acts as multiplication by a measurable function which takes S onto a reproducing kernel Hilbert space of functions which are analytic in a region containing \mathbbR{\mathbb{R}} , and are meromorphic in \mathbbC{\mathbb{C}} . In the process of establishing this result, several new results on the spectra and spectral representations of symmetric operators are proven. It is further observed that there is a large class of de Branges functions E, for which the de Branges spaces H(E) ì L²(\mathbbR, |E(x)|^-2dx){\mathcal{H}(E) \subset L^{2}(\mathbb{R}, |E(x)|^{-2}dx)} are examples of subspaces satisfying the conditions of this result.     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).