Localizable spectrum and bounded local resolvent functions

Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown that, for any measure with natural spectrum on a locally compact abelian group, the corresponding operator of convolution on the group algebra admits a non-zero bounded local resolvent function precisely when its spectrum has non-empty interior. Received: 15 November 2007     

Operators,absolutely indefinitely bounded below,in spaces with indefinite metric

Let r be the spectral radius of an operatorU, absolutely indefinitely bounded below. It is proved that r = c^1/, where c is the exact lower bound ofU and is a number occurring in the definition of the I-metric. A bound is obtained for the dimensionality of the direct sum of root lineals ofU (c 1), corresponding to eigenvalues whose absolute values are smaller than unity.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 301–305, September, 1971.     

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behavior of the s-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain Ω with smooth boundary ∂Ω. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on ∂Ω. It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order p for which

     

On a resolvent estimate of the interface problem for the Stokes system in a bounded domain

Obtained is the L_p estimate of solutions to the resolvent problem for the Stokes system with interface condition in a bounded domain in . It is the first step to consider the free boundary value problem.     

Counting cubic extensions with given quadratic resolvent

Given a number field k and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of k whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^{\alpha })$, for an explicit $\alpha <1$.     

Klein resolvent for an equation of fifth degree over a field of characteristic dividing into 5!

Klein's familiar constructions of the single-parameter resolvent for an equation of fifth degree cannot be extended without modification to fields of characteristic 2, 3, or 5. It is shown that, for such fields, the single-parameter resolvent exists, and its construction is described.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 46, pp. 36–40, 1974.     

Continuity of Operators Intertwining with Convolution Operators

Let G be a locally compact abelian group, let μ be a bounded complex-valued Borel measure on G, and let T_μ be the corresponding convolution operator on L¹(G). Let X be a Banach space and let S be a continuous linear operator on X. Then we show that every linear operator Φ: X→L¹(G) such that ΦS=T_μΦ is continuous if and only if the pair (S,T_μ) has no critical eigenvalue.