Showing papers in "Pacific Journal of Mathematics in 1969"

Isometries of certain function spaces

Abstract: Let X be a discrete symmetric Banach function space with absolutely continuous norm. We prove by the method of generalized hermitian operator that an operator U on X is an onto isometry if and only if it is of the form: Uf(.)=u(.)f(T.) all feX, where % is a unimodular function and T is a set isomorphism of the underlying measure space. That other types of isometries occur if the symmetry condition is not present is illustrated by an example. We completely describe the isometries of a reflexive Orlicz space LMΦ(Γ^LZ) provided the atoms have equal mass (the atom-free case has been treated by G. Lumer); similarly for the case that no Hubert subspace occurs.

Lifting projections of convex polyhedra.

Abstract: If τ is a projection of a closed convex polyhedron P onto a convex polyhedron Q, then a lifting of Q into P is defined to be a single-valued inverse τ* of τ such that τ*(Q) is the union of closed faces of P The main result of this paper, designated the Lifting Theorem, asserts that there always exists a lifting r*, provided only that there exists at least one face of P on which τ acts one-to-one The lifting theorem represents a unifying generalization of a number of results in the theory of convex polyhedra and should prove useful as an investigative as well as a conceptual tool In the course of the proof, a special case of the Lifting Theorem is translated into linear programming terms and stated as the Basis Decomposition Theorem, which summarizes the behavior of a linear program as a function of its right-hand sides In particular, the fact that a lifting is necessarily a piecewise linear homeomorphism is reflected in the Basis Decomposition Theorem as the observation that the optimal solution of a linear program can always be chosen as a continuous function of the right-hand sides

Cell-like mappings. II.

Abstract: Cell-like mappings are introduced and studied. A space is cell-like if it is homeomorphic to a cellular subset of some manifold. A mapping is cell-like if its point-inverses are celllike spaces. It is shown that proper, cell-like mappings of ENR'S (Euclidean NRs) form a category which includes both proper, contractible maps of ENR's and proper, cellular maps from manifolds to ENR's. It is difficult to break out of the category: The image of a proper, cell-like map on an ENR, is again an ENR, provided the image is finite-dimensional and Ήausdorff. Some applications to (unbounded) manifolds are given. For example: A cell-like map between topological manifolds of dimension ^ 5 is cellular. The property of being an open ncell, n ^ 5, is preserved under proper, cell-like maps between topological manifolds. The image of a proper, cellular map on an %-manifold is a homotopy %-manifold.

Approximation by inner functions

Abstract: Let L°°(T) denote the complex Banach algebra of (equivalence classes of) bounded measurable functions on the unit circle T, relative to Lebesgue measure m. The norm 11 /11» of an / in L^iT) is the essential supremum of |/| on T. The collection of all bounded holomorphic functions in the open unit disc U forms a Banach algebra which can be identified (via radial limits) with the norm-closed subalgebra H°° of L~(T). A function / in L°°(T) is unimodular if |/| = 1 a.e., on Tm The inner functions are the unimodular members of H°°. It is well known that they play an important role in the study of H°°. The main result (Theorem 1) is that the set of quotients of inner functions is norm-dense in the set of unimodular functions in L°°(T). One consequence of this (Theorem 7) is that the set of radial limits of holomorphic functions of bounded characteristic in U is norm-dense in L°°(T). It is also shown (Theorem 3, 4) that the Gelfand transforms of the inner functions separate points on the Silov boundary of H°°, and this is used to obtain a new proof (and generalization) of a theorem of D. J. Newman (Theorem 4).

Best constants in a class of integral inequalities

Abstract: \\y\\*\\y\\*w(x)dχ£Ki\\ \\ y \\m{x)dx\\ where yia) = y(ά) = = y-\\a) = 0 and y~ is absolutely continuous. It is first shown that for a certain class of m and w, equality can be attained in the inequality. Applying variational techniques reduces the determination of the best constant to a nonlinear eigenvalue problem for an integral operator. If m and w are sufficiently smooth this reduces further to a boundary value problem for a differential equation. The method is illustrated by determining the best constants in case (a, b) is a finite interval, mix) = wix) = 1, and n = 1.

Partial differential equations of Sobolev-Galpern type

Abstract: A mixed initial and boundary value problem is considered for a partial differential equation of the form Mut(x, t)+Lu(x, £)=0, where M and L are elliptic differential operators of orders 2 m and 21, respectively, with m ^ I. The existence and uniqueness of a strong solution of this equation in Hι0(G) is proved by semigroup methods.

On Ulam’s conjecture for separable graphs

Abstract: Ulam's conjecture, that every graph of order greater than two is determined up to isomorphism by its collection of maximal subgraphs, is verified for the case of separable graphs which have no pendant vertices. Partial results are then obtained for the case of graphs with pendant vertices. Unless otherwise stated, the graphs dealt with in this paper will be finite and undirected, and may have loops and multiple edges. Any definitions and notation not given below can be found in Berge [1]. A part Gι of a graph G is a subset of the vertices and edges of G. The end-vertices of edges in Gι need not themselves be in G\ If Gι is a part of G, G — G1 denotes that partial subgraph of G which is obtained by deleting G1 and all edges of G which are joined to vertices of G1. Now let S be some distinguished set of parts of a graph, and let S(X) — {X1} be the labelled set of these parts in the graph X We call two graphs G, H S-equίvalent if | S(G) \ = | S(H) | = M« oo) and, possibly after relabelling, G - Gι ^ H - if*(l ^ i ^ M). G\ Hι will be referred to as corresponding parts. In [8] Ulam proposed the following conjecture. CONJECTURE A. Vertex-equivalent graphs of order greater than two are isomorphic.

Vector valued Orlicz spaces generalized N-functions. I

Abstract: The theory of Orlicz spaces generated by JV-f unctions of a real variable is well known. On the other hand, as was pointed out by Wang, this same theory generated by ^-functions of more than one real variable has not been discussed in the literature. The purpose of this paper is to develop and study such a class of generalized iV-functions (called GNfunctions) which are a natural generalization of the functions studied by Wang and the variable iV-functions by Portnov. In second part of this study we will utilize GiV-functions to define vector-valued Orlicz spaces and examine the resulting theory.

Vector valued Orlicz spaces. II.

Abstract: In this paper properties of linear spaces generated by GNfunctions, which are called vector valued Orlicz spaces, are studied. The class of GΛΓ-functions were introduced and studied by the author in the paper Vector Valued Orlicz Spaces, I. This work extends the usual theory of Orlicz spaces generated by real valued ΛΓ-functions of a real variable. In particular, GN-functions are a generalization of the variable N-functions used by Portnov and the nondecreasing N-functions by Wang.

An embedding theorem for lattice-ordered fields

Abstract: In this paper we develop a method for constructing latticeordered fields (\"^-fields\") which are not totally ordered (\"ofields\") and hence are not /-rings. We show that many of these fields admit a Hahn type embedding into a field of formal power series with real coefficients. In order to establish such an embedding we make use of the valuation theory for abelian -S^-groups and prove the \"well known\" fact that each o-field can be embedded in an o-field of formal power series.