Simplicity of stable principal sheaves

Let M be a compact connected Kähler manifold, and let Gbe a connected complex reductive linear algebraic group. Weprove that a principal G-sheaf on M admits an admissible Einstein–Hermitianconnection if and only if the principal G-sheaf is polystable.Using this it is shown that the holomorphic sections of theadjoint vector bundle of a stable principal G-sheaf on M aregiven by the center of the Lie algebra of G. The Bogomolov inequalityis shown to be valid for polystable principal G-sheaves.     

A restriction theorem for oscillatory integral operator with certain polynomial phase

We consider the oscillatory integral operator \({T_{\alpha ,m}}f\left( x \right) = {\smallint _{{\mathbb{R}^n}}}{e^i}\left( {x_1^{{\alpha _1}}y_1^m + \cdots x_1^{{\alpha _n}}y_n^m} \right)f\left( y \right)dy\) where the function f is a Schwartz function. In this paper, the restriction theorem on S ^n-1 for this operator is obtained. Moreover, we obtain a necessary condition which ensures validity of the restriction theorem.     

On the stable principal Higgs sheaves

Let G be a connected reductive linear algebraic group defined over C with Lie algebra g. Let be a stable principal Higgs G-sheaf on a compact connected Kähler manifold. We consider all holomorphic sections of the adjoint vector bundle ad(EG) of EG that commute with the Higgs field φ. These correspond to the infinitesimal automorphisms of the principal Higgs G-sheaf. Any element of the center of g gives such a section. We prove that all the sections are given by the center of g.     

Character sheaves on certain spherical varieties

We study a class of perverse sheaves on some spherical varieties which include the strata of the De Concini-Procesi completion of a symmetric variety. This is a generalization of the theory of (parabolic) character sheaves.     

Equivalence of polynomial conjectures in additive combinatorics

We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for U ³, which relates to functions which locally look like quadratics. In both cases a weak form, with exponential decay of parameters is known, and a strong form with only a polynomial loss of parameters is conjectured. Our main result is that the two conjectures are in fact equivalent.     

Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval,for polynomial operator pencils

Under investigation is the equivalence of derived chains constructed from root vectors of polynomial pencils of operators acting in a Hilbert space. These derived chains correspond to various boundary—value problems on a finite interval for an operator—differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 83–95, January, 1990.     

Isometries of certain operator algebras

To a given basis on an -dimensional Hilbert space , we associate the algebra of all linear operators on having every as an eigenvector. So, is commutative, semisimple, and -dimensional. Given two algebras of this type, and , there is a natural algebraic isomorphism of and . We study the question: When does preserve the operator norm?

     

Isometries of certain operator spaces

Let and be Banach spaces, and be the spaces of bounded linear operators from into In this paper we give full characterization of isometric onto operators of for a certain class of Banach spaces, that includes We also characterize the isometric onto operators of and the compact operators on Furthermore, the multiplicative isometric onto operators of , when multiplication on is taken to be the Schur product, are characterized.

     

Multiplication operator on a matrix polynomial

It is proved that the study of a perturbed multiplication operator on a matrix polynomial in the space L₂(, ⁿ) may be reduced to the study of a perturbed multiplication operator with independent variable in the space L₂(, , ^N) with weight satisfying the Mackenhaupt condition.Translated from Ukrayins'kyy Matemarychnyy Zhurnal, Vol. 44, No. 9, pp. 1287–1289, September, 1992.     

Generalization of certain well-known polynomial inequalities

If is a polynomial of degree n having no zeros in |z|<1, then for |β|?1, it was proved by Jain [V.K. Jain, Generalization of certain well known inequalities for polynomials, Glas. Mat. 32 (52) (1997) 45-51] that

     

The polynomial dual of an operator ideal

We prove that the adjoint of a continuous homogeneous polynomial $P$ between Banach spaces belongs to a given operator ideal $\mathcal I$ if and only if $P$ admits a factorization $P = u \circ Q$ where the adjoint of the linear operator $u$ belongs to $\mathcal I$ . Several consequences of this factorization are obtained, for example we characterize the polynomials whose adjoints are absolutely $p$ -summing.     

The linear factorization of polynomial operator pencils

Sufficient conditions are found for the linear factorization of polynomial operator pencils of arbitrary order in a Banach space. This factorization is generated by the solution of an appropriate operator equation.Translated from Matematicheskie Zametki, Vol. 13, No. 4, pp. 551–559, April, 1973.The author wishes to express his thanks to A. G. Kostyuchenko for the formulation of the problem.