Other Mathematics Commons^™

Prolongations And Cyclic Vectors, William T. Ross, Harold S. Shapiro

Department of Math & Statistics Faculty Publications

For functions belonging to invariant subspaces of the backward shift operator Bf = (f − f(0))/z on spaces of analytic functions on the unit disk D, we explore, in a systematic way, the continuation properties of these functions.

Bergman Spaces On Disconnected Domains, William T. Ross, Alexandru Aleman, Stefan Richter

Department of Math & Statistics Faculty Publications

For a bounded region G C C and a compact set K C G, with area measure zero, we will characterize the invariant subspaces M (under f -> zf)of the Bergman space L^p_a(G \ K), 1 ≤ p < ∞, which contain L^p_a(G) and with dim(M/(z - λ)M) = 1 for all λϵ G \ K. When G \ K is connected, we will see that di\m(M /(z — λ)M) = 1 for all λ ϵ G \ K and thus in this case we will have a complete …

An Invariant Subspace Problem For P = 1 Bergman Spaces On Slit Domains, William T. Ross

Department of Math & Statistics Faculty Publications

In this paper, we characterize the z-invariant subspaces that lie between the Bergman spaces A¹(G) and A¹(G/K), where G is a bounded region in the complex plane and K is a compact subset of a simple arc of class C¹.

Weak-Star Limits On Polynomials And Their Derivatives, William T. Ross, Joseph A. Ball

Department of Math & Statistics Faculty Publications

Let μ and v be regular finite Borel measures with compact support in the real line ℝ and define the differential operator D :L ∞(μ) → L ∞(v) with domain equal to the polynomials P by Dp = p′. In this paper we will characterize the weak-star closure of the graph of D in ∞(μ) ⊕ ∞(y). As a consequence we will characterize when D is closable (i.e. the weak-star closure of G contains no non-zero elements of the form o ⊕ g) and when g is weak-star dense in L∞(μ) ⊕ …

Partially Confluent Maps And N-Ods, Van C. Nall

Department of Math & Statistics Faculty Publications

Let f : X-->Y be a map between topological spaces. A Wf-set in Y is a continuum in Y which is the image under f of a continuum in X. The map f is partially confluent if each continuum in Y is the union of a finite number of Wf-sets, and n-partially confluent if each continuum in Y is the union of n Wf-sets. In this paper, it is shown that every partially confluent map onto an n-cell is weakly confluent. Also, the relationship between partially confluent maps and continua which …

Maps Which Preserve Graphs, Van C. Nall

Department of Math & Statistics Faculty Publications

In 1976 Eberhart, Fúgate, and Gordh proved that the weakly confluent image of a graph is a graph. A much weaker condition on the map is introduced called partial confluence, and it is shown that the image of a graph is a graph if and only if the map is partially confluent.

In addition, it is shown that certain properties of one-dimensional continua are preserved by partially confluent maps, generalizing theorems of Cook and Lelek, Tymchatyn and Lelek, and Grace and Vought. Also, some continua in addition to graphs are shown to be the images of partially confluent maps only.

Weak Confluence And W-Sets, Van C. Nall

Department of Math & Statistics Faculty Publications

A mapping between continua is weakly confluent if for each subcontinuum K of the range some component of the preimage of K maps onto K. Class [W] is the class of all continua which are the images of weakly confluent maps only. The notion of Class [W] was introduced by Andrej Lelek in 1972. Since then it has been widely explored and some characterizations of these continua have been given. J. Grispolakis and E. D. Tymchatyn have given a characterization in terms of hyperspaces [4]. J. Davis has shown that acyclic atriodic continua are in Class [W]i therefore, atriodic tree-like …