Inequalities for quadratic polynomials in Hermitian and dissipative operators

Let T be a Hermitian operator on a Banach space and let P be a real quadratic polynomial. Among other inequalities we give lower bounds for |P(T)x| in terms of |x|, |Tx|, and |T²x|. As a special case we deduce extensions of some classical inequalities involving derivatives of a function and obtain some new inequalities of this kind.     

On commutative semigroups of polynomials with respect to composition

In the first part, commutative semigroups of Dickson polynomials with respect to composition over integral domains are studied. Subsequently we investigate commutative semigroups of polynomials with respect to composition over fields or integral domains. all such semigroups which contain only polynomials of positive even degree and at least one polynomial of each positive even degree over a field of characteristic 2 and 3 are determined. Further, all commutative semigroups which contain only polynomials of odd degree and at least one polynomial of each odd degree over fields of characteristic 3 and 5 are characterised. Where appropriate the corresponding results for polynomials over integral domains are given.Gewidmet unserem hochverehrten Lehrer E. Hlawka zum siebzigsten GeburtstagThe first author obtained support from ARGS grant F 84/15183. The research of the second author was partially supported by the Österreichischen Fonds zur Förderung der wissenschaftlichen Forschung under FWF-Project Nr 5452. This latter author also wants to express his thanks to the members of the Mathematics Department at Monash University, Clayton, australia, for their support and hospitality during a visiting appointment of 3 months in 1985, when this paper was written.     

Compact Hermitian surfaces of Einstein type with respect to the Hermitian connection

Two Einstein-type conditions for the Hermitian curvature tensor are considered on a compact Hermitian surface: It is proved that if the symmetric part of the Ricci tensors is a scalar multiple of the metric with a negative constant, then the metric is Kaehler. If the Hermitian surface satisfies the Hermite-Einstein condition with a non positive constant, then the metric is Kaehler.Supported by Contract MM 413/1994 with the Ministry of Science and Education of Bulgaria.Supported by Contract MM 423/1994 with the Ministry of Science and Education of Bulgaria and by Contract 219/1994 with the University of Sofia St. Kl. Ohridski.     

对角无穷维Hamilton算子点谱关于实轴的对称性

在不同条件下得到对角无穷维Hamilton算子点谱的两个组成部分σp（A）与σp（-A^＊）关于实轴对称的充分必要条件.以此为基础,完全刻画了对角无穷维Hamilton算子点谱关于实轴的对称性.     

On polynomials hypoelliptic with respect to a group of variables

For a class of polynomials a necessary and sufficient condition is found for those polynomials to be hypoelliptic with respect to a group of variables.     

On polynomials orthogonal with respect to particular variable-signed weight functions

In this paper we examine the existence of polynomials orthogonal with respect to weight functions which are given by the product of a nonnegative (on the interval of orthogonality) function and a polynomial. In particular, we give some results when the polynomial factor is orthogonal with respect to the remaining (nonnegative) part of the weight function.

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Sommario In questo lavoro esaminiamo l'esistenza di polinomi ortogonali rispetto a funzioni peso date dal prodotto di una funzione non negativa (nell' intervallo di ortogonalità) per un polinomio. In particolare diamo alcuni risultati quando il fattore polinomiale è ortogonale rispetto alla rimanente parte (non negativa) della funzione peso.

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Work performed under the auspices of the Italian Research Council.     

On products of two Hermitian operators

In this paper, we study the problem of characterizing the bounded linear operators on a Hilbert space that admit a factorization as a product of two Hermitian operators. It is shown that a normal operator can be decomposed as a product of two Hermitian operators if and only if it is similar to its adjoint. Some partial results about hyponormal operators are obtained.     

On the spectrum of elliptic operators with respect to indefinite weights

Recent results on linear elliptic eigenvalue problems with respect to indefinite weight functions are recalled, and some applications are given.     

On derivatives of hypergeometric functions and classical polynomials with respect to parameters

Closed expressions are obtained for derivatives of symbolic order with respect to parameters for the hypergeometric functions, Laguerre, Gegenbauer, Jacobi and some other polynomial.     

On Markov operators preserving polynomials

The paper is concerned with a special class of positive linear operators acting on the space C(K)$C(K)$ of all continuous functions defined on a convex compact subset K   of R^d${\mathbf{R}}^d$, d?1$d?1$, having non-empty interior. Actually, this class consists of all positive linear operators T   on C(K)$C(K)$ which leave invariant the polynomials of degree at most 1 and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if ∂K   is an ellipsoid. Furthermore, a characterization of balls of R^d${\mathbf{R}}^d$ in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.     

Commuting difference operators, spinor bundles and the asymptotics of orthogonal polynomials with respect to varying complex weights

The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.In the second part we show that given an arbitrary nodal hyperelliptic curve satisfying certain conditions of admissibility we can reconstruct a sequence of polynomials orthogonal with respect to semiclassical complex varying weights supported on several curves in the complex plane. The strong asymptotics of these polynomials will be shown to be given by the spinors introduced in the first part using a Riemann-Hilbert analysis.In the third part we use Strebel theory of quadratic differentials and the procedure of welding to reconstruct arbitrary admissible hyperelliptic curves. As a result we can obtain orthogonal polynomials whose zeroes may become dense on a collection of Jordan arcs forming an arbitrary forest of trivalent loop-free trees.     

On operators with the same local spectra

Let B(X) be the algebra of all bounded linear operators in a complex Banach space X. We consider operators T ₁, T ₂ B(X) satisfying the relation for any vector x X, where _T (x) denotes the local spectrum of T B(X) at the point x X. We say then that T ₁ and T ₂ have the same local spectra. We prove that then, under some conditions, T ₁ – T ₂ is a quasinilpotent operator, that is as n . Without these conditions, we describe the operators with the same local spectra only in some particular cases.     

On Fourier series with respect to the Walsh-Kaczmarz system

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