Hardy-type spaces for eigenfunctions of invariant differential operators on homogeneous line bundles over Hermitian symmetric spaces

We prove a Fatou-type theorem on a homogeneous line bundle over a Hermitian symmetric space, and characterize the range of the Poisson transform of L_p -functions on the maximal boundary as a Hardy-type space.     

Darboux transformations and recursion operators for differential-difference equations

We review two concepts directly related to the Lax representations of integrable systems: Darboux transformations and recursion operators. We present an extensive list of integrable differential-difference equations with their Hamiltonian structures, recursion operators, nontrivial generalized symmetries, and Darboux-Lax representations. The new results include multi-Hamiltonian structures and recursion operators for integrable Volterra-type equations and integrable discretizations of derivative nonlinear Schrödinger equations such as the Kaup-Newell, Chen-Lee-Liu, and Ablowitz-Ramani-Segur (Gerdjikov-Ivanov) lattices. We also compute the weakly nonlocal inverse recursion operators.     

Completely integrable equations on homogeneous spaces

Completely integrable linear Pfaff systems are investigated, and some of their generalizations to manifolds M=G/, where G is a Lie group and is a discrete subgroup of G, are studied. The reducibility of such a system to a system with constant coefficients with respect to a natural parallelism on M is considered.Translated from Matematicheskie Zametki, Vol. 9, No. 4, pp. 365–373, April, 1971.     

Generating operators for integrable nonlinear evolution equations

For linear problems which are associated with known, exactly integrable nonlinear evolution equations, one gives the corresponding integrodifferential Λ-operators. Relative to the expansions with respect to the elgenfunctions of Λ-operators, the method of the inverse scattering problem can be considered as the analog of the Fourier transform of linear problems, while the Λ-operators are the analogues of the differentiation operator. One considers the equations: Koteweg-de Vries, the nonlinear Schrödinger equations, the nonlinear Schrödinger equations with a derivative, the system of three waves, the matricial analog of the KdV equation, the Toda chain equation.     

Graded differential equations and their deformations: A computational theory for recursion operators

An algebraic model for nonlinear partial differential equations (PDE) in the category ofn-graded modules is constructed. Based on the notion of the graded Frölicher-Nijenhuis bracket, cohomological invariants H ^* (A) are related to each object (A, ) of the theory. Within this framework, H ⁰ (A) generalizes the Lie algebra of symmetries for PDE's, while H ¹ (A) are identified with equivalence classes of infinitesimal deformations. It is shown that elements of a certain part of H ¹ (A) can be interpreted as recursion operators for the object (A, ), i.e. operators giving rise to infinite series of symmetries. Explicit formulas for computing recursion operators are deduced. The general theory is illustrated by a particular example of a graded differential equation, i.e. the Super KdV equation.Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia.     

Symplectic Dirac operators on Hermitian symmetric spaces

We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.     

Rational curves on moduli spaces of vector bundles

We identify the spaces Hom_i(ℙ¹,M) fori = 1, 2, whereM is the moduli space of vector bundles of rank 2 and determinant isomorphic to ,x ₀ ∈X, on a compact Riemann surface of genusg ≥ 2.     

Isometries and Hermitian operators on complex symmetric sequence spaces

We consider a complex symmetric sequence space E that possesses the Fatou property and is different from l₂. We prove that, for every surjective linear isometry V on E, there exist λ _n ∈ ? with |λ _n | = 1 and a bijective mapping π on the set ? of natural numbers such that

$$V\left( {\left\{ {\xi _n } \right\}_{n \in \mathbb{N}} } \right) = \left\{ {\lambda _n \xi _{\pi (n)} } \right\}_{n \in \mathbb{N}}$$

for every {ξ _n {_n∈? ∈ E.

     

Completely positive compact operators on non-commutative symmetric spaces

Under natural conditions, it is shown that a completely positive operator between two non-commutative symmetric spaces of τ-measurable operators which is dominated in the sense of complete positivity by a completely positive compact operator is itself compact.     

Trace formulas for nuclear operators in spaces of Bochner integrable functions

The paper is devoted to trace formulas for nuclear operators in spaces of Bochner integrable functions. We characterise nuclearity for integral operators on such spaces and develop a trace formula for general kernels applying vector-valued maximal functions.     

A note on the surjectivity of operators on vector bundles over discrete spaces

In this note we give a short and self-contained proof for a criterion of Eidelheit on the solvability of linear equations in infinitely many variables. We use this criterion to study the surjectivity of magnetic Schrödinger operators on bundles over graphs.     

Invariant differential operators on Hermitian symmetric spaces and their eigenvalues

Let be the invariant Cauchy Riemann operator and the corresponding invariant Laplacians on a bounded symmetric domain. We calculate the eigenvalues ofM _m on spherical functions. In particular we prove that for a symmetric domain of rank two the operatorsM ₁,M ₃ generate all invariant differential operators. We also find the eigenvalues of the generators introduced by Shimura.     

Norms of multiplication operators on Hardy spaces and weighted composition operators from Hardy spaces to weighted-type spaces on bounded symmetric domains

Let D be a bounded symmetric domain. We calculate operator norm of the multiplication operator on the Hardy space H^p(D), as well as of the weighted composition operator from H^p(D) to a weighted-type space.     

Rational surfaces and moduli spaces of vector bundles on rational surfaces

Let X be a smooth algebraic surface, L ? Pic(X) L \in \textrm{Pic}(X) and H an ample divisor on X. Set M_X,H(2; L, c₂) the moduli space of rank 2, H-stable vector bundles F on X with det(F) = L and c₂(F) = c₂. In this paper, we show that the geometry of X and of M_X,H(2; L, c₂) are closely related. More precisely, we prove that for any ample divisor H on X and any L ? Pic(X) L \in \textrm{Pic}(X) , there exists n₀ ? \mathbbZ n_0 \in \mathbb{Z} such that for all n₀ \leqq c₂ ? \mathbbZ n_0 \leqq c_2 \in \mathbb{Z} , M_X,H(2; L, c₂) is rational if and only if X is rational.     

Nonlocal symmetries and recursion operators: Partial differential and differential-difference equations

A systematic method to derive the nonlocal symmetries for partial differential and differential-difference equations with two independent variables is presented and shown that the Korteweg-de Vries (KdV) and Burger's equations, Volterra and relativistic Toda (RT) lattice equations admit a sequence of nonlocal symmetries. An algorithm, exploiting the obtained nonlocal symmetries, is proposed to derive recursion operators involving nonlocal variables and illustrated it for the KdV and Burger's equations, Volterra and RT lattice equations and shown that the former three equations admit factorisable recursion operators while the RT lattice equation possesses (2×2) matrix factorisable recursion operator. The existence of nonlocal symmetries and the corresponding recursion operator of partial differential and differential-difference equations does not always determine their mathematical structures, for example, bi-Hamiltonian representation.     

Ramadanov conjecture and line bundles over compact Hermitian symmetric spaces

We compute the Szegö kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in ${\mathbb C^n}We compute the Szeg? kernels of the unit circle bundles of homogeneous negative line bundles over a compact Hermitian symmetric space. We prove that their logarithmic terms vanish in all cases and, further, that the circle bundles are not diffeomorphic to the unit sphere in \mathbb Cⁿ{\mathbb C^n} for Grassmannian manifolds of higher ranks. In particular, they provide an infinite family of smoothly bounded strictly pseudoconvex domains on complex manifolds for which the logarithmic term in the Fefferman expansion of the Szeg? kernel vanishes but whose boundary is not diffeomorphic to the sphere (in fact, it is not even locally spherical). The analogous results for the Bergman kernel are also obtained.