Sufficient conditions for one domain to contain another in a space of constant curvature

As an application of the analogue of C-S. Chen's kinematic formula in the 3-dimensional space of constant curvature , that is, Euclidean space , -sphere , hyperbolic space (, respectively), we obtain sufficient conditions for one domain to contain another domain in either an Euclidean space , or a -sphere or a hyperbolic space .

     

On uniqueness of -adic meromorphic functions

Let be a complete ultrametric algebraically closed field of characteristic zero, and let be the field of meromorphic functions in . For all set in and for all we denote by the subset of : zero of order After studying unique range sets for entire functions in in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every , there exist sets of elements in such that, if have the same poles (counting multiplicities), and satisfy , then . We show how to construct such sets.

     

A topology on lattice ordered groups

We show that a lattice ordered group can be topologized in a natural way. The topology depends on the choice of a set of admissible elements (-topology). If a lattice ordered group is 2-divisible and satisfies a version of Archimedes' axiom (-group), then we show that the -topology is Hausdorff. Moreover, we show that a -group with the -topology is a topological group.

     

Regularization of semigroups that are strongly continuous for

Let be a Banach space and a strongly continuous semigroup with . We show that the generator of generates a regularized semigroup. Our construction of a regularizing operator uses an existence result of J. Esterle.

     

The maximal normal -group

Let be a prime number and let be an abelian -group. Let be the maximal normal -subgroup of and the maximal -subgroup of its centre. Let be the torsion radical of . Then . The result is new for and 3, and the proof is new and valid for all primes .

     

On complementary subspaces of Hilbert space

Every pair of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form on a Hilbert space . Here is possibly , is a positive injective contraction and denotes the graph of . For such a pair the following are equivalent: (i) is similar to a pair in generic position; (ii) and have a common algebraic complement; (iii) is similar to for some operators on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.

     

Differential forms on quotients by reductive group actions

Let be a smooth affine algebraic variety where a reductive algebraic group acts with a smooth quotient space . We show that the algebraic differential forms on which are pull-backs of forms on are exactly the -invariant horizontal differential forms on .

     

Bourgain's analytic projection revisited

For a positive function on the unit circle with , the following two statements are equivalent: (a) ; (b) there is an operator projecting onto for all at once and having weak type (1,1) with respect to .

     

The exposed points of the set of invariant means on an ideal

Let be a -compact locally compact nondiscrete group and let be a -invariant ideal of . We denote the set of left invariant means on that are zero on (i.e. for all ) by . We show that, when is amenable as a discrete group and the closed -invariant subset of the spectrum of corresponding to is a -set, is very large in the sense that every nonempty -subset of contains a norm discrete copy of , where is the Stone- compactification of the set of positive integers with the discrete topology. In particular, we prove that has no exposed points in this case and every nonempty -subset of the set of left invariant means on contains a norm discrete copy of .

     

Continuity of Lie mappings of the skew elements of Banach algebras with involution

Let and be centrally closed prime complex Banach algebras with linear involution. If is semisimple, then any Lie derivation of the skew elements of is continuous and any Lie isomorphism from the skew elements of onto the skew elements of is continuous.

     

Global iteration schemes for strongly pseudo-contractive maps

Suppose is a real uniformly smooth Banach space, is a nonempty closed convex and bounded subset of , and is a strong pseudo-contraction. It is proved that if has a fixed point in then both the Mann and the Ishikawa iteration processes, for an arbitrary initial vector in , converge strongly to the unique fixed . No continuity assumption is necessary for this convergence. Moreover, our iteration parameters are independent of the geometry of the underlying Banach space and of any property of the operator.

     

-ideals of compact operators are separably determined

We prove that the space of compact operators on a Banach space is an -ideal in the space of bounded operators if and only if has the metric compact approximation property (MCAP), and is an -ideal in for all separable subspaces of having the MCAP. It follows that the Kalton-Werner theorem characterizing -ideals of compact operators on separable Banach spaces is also valid for non-separable spaces: for a Banach space is an -ideal in if and only if has the MCAP, contains no subspace isomorphic to and has property It also follows that is an -ideal in for all Banach spaces if and only if has the MCAP, and is an -ideal in .

     

Character degrees and local subgroups of -separable groups

Let be a finite -solvable group for different primes and . Let and be such that . We prove that every of -degree has -degree if and only if and .

     

Mergelyan pairs for harmonic functions

Let be open and be a bounded set which is closed relative to . We characterize those pairs such that, for each harmonic function on which is uniformly continuous on , there is a sequence of harmonic polynomials which converges to uniformly on . As an immediate corollary we obtain a characterization of Mergelyan pairs for harmonic functions.

     

A unified extension of two results of Ky Fan on the sum of matrices

Let be an Hermitian matrix with where are the ordered eigenvalues of . A result of Ky Fan (1949) asserts that if and are Hermitian matrices, then is majorized by . We extend the result in the framework of real semisimple Lie algebras in the following way. Let be a noncompact real semisimple Lie algebra with Cartan decomposition . We show that for any given , , where is the unique element corresponding to , in a fixed closed positive Weyl chamber of a maximal abelian subalgebra of in . Here the ordering is induced by the dual cone of . Fan's result corresponds to the Lie algebra . The compact case is also discussed. As applications, two unexpected singular values inequalities concerning the sum of two real matrices and the sum of two real skew symmetric matrices are obtained.

     

On an analogue of Selberg's eigenvalue conjecture for

Let be the homogeneous space associated to the group
. Let where and consider the first nontrivial eigenvalue of the Laplacian on . Using geometric considerations, we prove the inequality . Since the continuous spectrum is represented by the band , our bound on can be viewed as an analogue of Selberg's eigenvalue conjecture for quotients of the hyperbolic half space.

     

Cycle rank of Lyapunov graphs and the genera of manifolds

We show that the cycle-rank of a Lyapunov graph on a manifold satisfies: , where is the genus of . This generalizes a theorem of Franks. We also show that given any integer with , for some Lyapunov graph on .

     

On the relative commutants of subfactors

Let be factors generated by a periodic tower of finite dimensional -algebras. We prove that for sufficiently large , is -isomorphic to a subalgebra of .

     

A note on invariance of spectrum for symmetric banach -algebras

Let be a symmetric Banach -algebra, let be a Banach algebra, and assume that . A result is proved giving conditions which imply that every element of has the same spectrum in both and .

     

Smooth rank one perturbations of selfadjoint operators

Let be a selfadjoint operator in a Hilbert space with inner product . The rank one perturbations of have the form , , for some element . In this paper we consider smooth perturbations, i.e. we consider for some . Function-theoretic properties of their so-called -functions and operator-theoretic consequences will be studied.