Let denote the spectral radius of an operator . We construct operators and on such that is discontinuous almost everywhere on the unit disk.
Let , where and is a Banach space. Let be an extension of to all of (i.e., ) such that has minimal (operator) norm. In this paper we show in particular that, in the case and the field is R, there exists a rank- such that for all if and only if the unit ball of is either not smooth or not strictly convex. In this case we show, furthermore, that, for some , there exists a choice of basis such that ; i.e., each is a Hahn-Banach extension of .
A new construction of semi-free actions on Menger manifolds is presented. As an application we prove a theorem about simultaneous coexistence of countably many semi-free actions of compact metric zero-dimensional groups with the prescribed fixed-point sets: Let be a compact metric zero-dimensional group, represented as the direct product of subgroups , a -manifold and (resp., ) its pseudo-interior (resp., pseudo-boundary). Then, given closed subsets of , there exists a -action on such that (1) and are invariant subsets of ; and (2) each is the fixed point set of any element .
Let , a prime (resp. , act freely on a finitistic space with (resp. rational) cohomology ring isomorphic to that of . In this paper we determine the possible cohomology algebra of the orbit space .
A variety is a class of Banach algebras , for which there exists a family of laws such that is precisely the class of all Banach algebras which satisfies all of the laws (i.e. for all , . We say that is an -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras , for which there exists a family of homogeneous laws such that is precisely the class of all Banach algebras , for which there exists 0$"> such that for all homogeneous polynomials , , where . However, there is no variety between the variety of all -algebras and the variety of all -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.
In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form
Here 0.$">
We prove that if m \ge 1,$"> then for any 0$"> there are choices of initial data from the energy space with initial energy such that the solution blows up in finite time. If we replace by , where is a sufficiently slowly decreasing function, an analogous result holds.
We prove that a Banach space has the compact range property (CRP) if and only if, for any given -algebra , every absolutely summing operator from into is compact. Related results for -summing operators () are also discussed as well as operators on non-commutative -spaces and -summing operators.
Let be a positive matrix-valued measure on a locally compact abelian group such that is the identity matrix. We give a necessary and sufficient condition on for the absence of a bounded non-constant matrix-valued function on satisfying the convolution equation . This extends Choquet and Deny's theorem for real-valued functions on .
We prove the existence of invariant projections from the Banach space of -pseudomeasures onto with for closed neutral subgroup of a locally compact group . As a main application we obtain that every closed neutral subgroup is a set of -synthesis in and in fact locally -Ditkin in . We also obtain an extension theorem concerning the Fourier algebra.
Every system of linearly independent homogeneous linear equations in unknowns with coefficients in has a unique (up to multiplication by ) non-zero solution vector in which the 's are integers with no common divisor greater than 1. It is known that, for large , can be arbitrarily greater than . We prove that if every equation, written as , is such that and are intervals of contiguous indices, then . This confirms conjectures of the author and Fred Roberts that arose in the theory of unique finite measurement.
The following dichotomy is established for any pair , of hereditary families of finite subsets of : Given , an infinite subset of , there exists an infinite subset of so that either , or , where denotes the set of all finite subsets of .
Let be a local ring and let be an ideal of positive height. If is a reduction of , then the coefficient ideal is by definition the largest ideal such that . In this article we study the ideal when the Rees algebra is Cohen-Macaulay.
Let be an open set and let denote the class of real analytic functions on . It is proved that for every surjective linear partial differential operator and every family depending holomorphically on there is a solution family depending on in the same way such that The result is a consequence of a characterization of Fréchet spaces such that the class of ``weakly' real analytic -valued functions coincides with the analogous class defined via Taylor series. An example shows that the analogous assertions need not be valid if is replaced by another set.
We estimate double exponential sums of the form
where is of multiplicative order modulo the prime and and are arbitrary subsets of the residue ring modulo . In the special case , our bound is nontrivial for with any fixed 0$">, while if in addition we have it is nontrivial for .