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.     

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.     

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.     

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.     

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.     

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.