复对称算子的一些等价性质

证明了在复对称算子的前提下,对数-亚正规算子与正规算子是等价的,并且给出了复对称算子的一些等价性质;最后通过给出例子来说明我们的结论.     

Differential symmetry breaking operators: I. General theory and F-method

We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method (F-method) based on the algebraic Fourier transform for generalized Verma modules, which characterizes differential symmetry breaking operators by means of certain systems of partial differential equations. In contrast to the setting of real flag varieties, continuous symmetry breaking operators of Hermitian symmetric spaces are proved to be differential operators in the holomorphic setting. In this case, symmetry breaking operators are characterized by differential equations of second order via the F-method.     

Green's Function of a Hartree-Type Equation with a Quadratic Potential

Using the complex WKB–Maslov method, we consider a solution of the Cauchy problem for a Hartree-type equation with a quadratic potential in the class of semiclassically supported functions. In this class, we obtain the evolution operator explicitly. We find parametric families of symmetry operators of the Hartree-type equation. Using the symmetry operators, we construct a family of exact solutions of this equation.     

Spectral triples and manifolds with boundary

We investigate manifolds with boundary in noncommutative geometry. Spectral triples associated to a symmetric differential operator and a local boundary condition are constructed. We show that there is no tadpole for classical Dirac operators with a chiral boundary condition on spin manifolds.     

Higher Laplace–Beltrami Operators on Bounded Symmetric Domains

It was conjectured by the first author and Peetre that the higher Laplace–Beltrami operators generate the whole ring of invariant operators on bounded symmetric domains. We give a proof of the conjecture for domains of rank ≤ 6 by using a graph manipulation of K¨ahler curvature tensor. We also compute higher order terms in the asymptotic expansions of the Bergman kernels and the Berezin transform on bounded symmetric domain.     

Vertex Operators for Standard Bases of the Symmetric Functions

We present formulas for operators which add a row or a column to the partition indexing the power, monomial, forgotten, Schur, homogeneous and elementary symmetric functions. As an application of these operators we show that the operator that adds a column to the Schur unctions can be used to calculate a formula for the number of pairs of standard tableaux the same shape and height less than or equal to a fixed k.     

Dissipative extension theory for linear relations

This work is devoted to dissipative extension theory for dissipative linear relations. We give a self-consistent theory of extensions by generalizing the theory on symmetric extensions of symmetric operators. Several results on the properties of dissipative relations are proven. Finally, we deal with the spectral properties of dissipative extensions of dissipative relations and provide results concerning particular realizations of this general setting.     

COMPLEX SYMMETRIC TOEPLITZ OPERATORS ON THE UNIT POLYDISK AND THE UNIT BALL

In this article, we study complex symmetric Toeplitz operators on the Bergman space and the pluriharmonic Bergman space in several variables. Surprisingly, the necessary and sufficient conditions for Toeplitz operators to be complex symmetric on these two spaces with certain conjugations are just the same. Also, some interesting symmetry properties of complex symmetric Toeplitz operators are obtained.     

Symmetry properties for solutions of nonlocal equations involving nonlinear operators

We pursue the study of one-dimensional symmetry of solutions to nonlinear equations involving nonlocal operators. We consider a vast class of nonlinear operators and in a particular case it covers the fractional p-Laplacian operator. Just like the classical De Giorgi's conjecture, we establish a Poincaré inequality and a linear Liouville theorem to provide two different proofs of the one-dimensional symmetry results in two dimensions. Both approaches are of independent interests. In addition, we provide certain energy estimates for layer solutions and Liouville theorems for stable solutions. Most of the methods and ideas applied in the current article are applicable to nonlocal operators with general kernels where the famous extension problem, given by Caffarelli and Silvestre, is not necessarily known.     

完备赋准范空间上一类无界线性随机算子的谱分解定理

本文利用随机内积空间方法给出了完备赋准范空间上一类无界线性随机算子的谱分解定理。此结果不仅推广了对称随机线性算子的谱分解定理，而且限于原情形也使其处理简明、清晰。     

On compact sets with a certain affine invariant

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.     

Two-dimensional systems that arise from the Noether classification of Lagrangians on the line

Noether-like operators play an essential role in writing down the first integrals for Euler-Lagrange systems of ordinary differential equations (ODEs). The classification of such operators is carried out with the help of analytic continuation of Lagrangians on the line. We obtain the classification of 5, 6 and 9 Noether-like operators for two-dimensional Lagrangian systems that arise from the submaximal and maximal dimensional Noether point symmetry classification of Lagrangians on the line. Cases in which the Noether-like operators are also Noether point symmetries for the systems of two ODEs are mentioned. In particular, the 8-dimensional maximal Noether algebra is remarkably obtained for the simplest system of the free particle equations in two dimensions from the 5-dimensional complex Noether algebra of the standard Lagrangian of the scalar free particle equation. We present the effectiveness of Noether-like operators for the determination of first integrals of systems of two nonlinear differential equations which arise from scalar complex Euler-Lagrange ODEs that admit Noether symmetry.     

Uniform primeness of the Jordan algebra of symmetric operators

In this note we establish the best possible constant for the general lower estimate for the Jacobson - McCrimmon operator on the algebra of symmetric operators acting on a Hilbert space.

     

Interpolation on Symmetric Spaces Via the Generalized Polar Decomposition

We construct interpolation operators for functions taking values in a symmetric space—a smooth manifold with an inversion symmetry about every point. Key to our construction is the observation that every symmetric space can be realized as a homogeneous space whose cosets have canonical representatives by virtue of the generalized polar decomposition—a generalization of the well-known factorization of a real nonsingular matrix into the product of a symmetric positive-definite matrix times an orthogonal matrix. By interpolating these canonical coset representatives, we derive a family of structure-preserving interpolation operators for symmetric space-valued functions. As applications, we construct interpolation operators for the space of Lorentzian metrics, the space of symmetric positive-definite matrices, and the Grassmannian. In the case of Lorentzian metrics, our interpolation operators provide a family of finite elements for numerical relativity that are frame-invariant and have signature which is guaranteed to be Lorentzian pointwise. We illustrate their potential utility by interpolating the Schwarzschild metric numerically.     

Differential symmetry breaking operators: II. Rankin–Cohen operators for symmetric pairs

Rankin–Cohen brackets are symmetry breaking operators for the tensor product of two holomorphic discrete series representations of \(SL(2,\mathbb {R})\). We address a general problem to find explicit formulæ  for such intertwining operators in the setting of multiplicity-free branching laws for reductive symmetric pairs. For this purpose, we use a new method (F-method) developed in Kobayashi and Pevzner (Sel. Math. New Ser., (2015). doi: 10.1007/s00029-15-0207-9) and based on the algebraic Fourier transform for generalized Verma modules.The method characterizes symmetry breaking operators by means of certain systems of partial differential equations of second order. We discover explicit formulæ  of new differential symmetry breaking operators for all the six different complex geometries arising from semisimple symmetric pairs of split rank one and reveal an intrinsic reason why the coefficients of orthogonal polynomials appear in these operators (Rankin–Cohen type) in the three geometries and why normal derivatives are symmetry breaking operators in the other three cases. Further, we analyze a new phenomenon that the multiplicities in the branching laws of Verma modules may jump up at singular parameters.     

Reflection symmetry and symmetrizability of Hilbert space operators

A general factorization theorem for symmetrizable operators relating their spectra to spectra of selfadjoint operators induced by minimal factorizations is established. Its modified version essentially improves and completes a theorem of Jorgensen, which concerns diagonalizing operators with reflection symmetry.

     

Hardy空间上一类$m$-复对称Toeplitz算子

本文主要研究Toeplitz算子相对于一对置换的共轭算子是2-复对称的充要条件. 首先在经典的Hardy空间上介绍一类被称为一对置换的共轭算子, 其次完整地刻画了在这类共轭算子下Toeplitz算子是2-复对称的结构, 利用Toeplitz算子在Hardy空间的经典正规正交基下的矩阵表示来刻画2-复对称Toeplitz算子. 最后对于Toeplitz算子分别补充前提$f_n=-f_{-n}$和$f_n=f_{-n}$, 得到了更简化的结果. 在第二个前提下, 研究Toeplitz算子的3-复对称性, 得到$T_f$关于$C_{(i,j)}$是3-CSO的结果和是2-CSO相同.     

On the Geometry of the Orbits of Hermann Actions

We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes of the orbits and a new proof of a Weyl-type integration formula for Hermann actions.      

四阶正则微分算子耦合自共轭边界条件的基本标准型

本文利用新的方法给出了4阶正则微分算子耦合自共轭边界条件的基本标准型,新标准型中的4个分块小矩阵为对称矩阵,且其行列式的模为1.这与2阶微分算子耦合边界条件的标准型极为类似,这为给出一般的高阶微分算子自共轭边界条件标准型提供了新的思路.     

正则和奇异自共轭微分算子的变分公式化(英文)

1 IntroductionVariational methods offer an appealing approach to the analysis and com-putations associated with symmetric differential equations because they areclose1y connected to the under1ying physica1 principle of energy. Such methodshave been used to obtain the existence and uniqueness results, to investigateproperties of solutions [2], I61, [5], I7], and to compute solutions as welI as ei-genvalues [11, [41 fOr linear and nonlinear equations. The first step in applyinga variational me…