In this paper, we prove some multiplicity results for sign-changing solutions of an operator equation in an ordered Banach space. The methods to show the main results of the paper are to associate a fixed point index with a strict upper or lower solution. The results can be applied to a wide variety of boundary value problems to obtain multiplicity results for sign-changing solutions.

The authors obtain some multiplier theorems on $H^p$ spaces analogous to the classical $L^p$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${\left(Tf\right)}^{(}x)=\lambda \left(x\right)f̂\left(x\right)$$(\lambda \in C(...$

We generalize some technical results of Glicksberg to the realm of general operator algebras and use them to give a characterization of open and closed projections in terms of certain multiplier algebras. This generalizes a theorem of J. Wells characterizing an important class of ideals in uniform algebras. The difficult implication in our main theorem is that if a projection is open in an operator algebra, then the multiplier algebra of the associated hereditary subalgebra arises as the closure...

We study multipliers M (bounded operators commuting with translations) on weighted spaces L ω p (ℝ), and establish the existence of a symbol µM for M, and some spectral results for translations S t and multipliers. We also study operators T on the weighted space L ω p (ℝ+) commuting either with the right translations S t , t ∈ ℝ+, or left translations P +S −t , t ∈ ℝ+, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S t ) of the operator S t proving that...

In 1966 de Branges and Rovnyak introduced a concept of complementation associated to a contraction between Hilbert spaces that generalizes the classical concept of orthogonal complement. When applied to Toeplitz operators on the Hardy space of the disc, H2, this notion turned out to be the starting point of a beautiful subject, with many applications to function theory. The work has been in constant progress for the last few years. We study here the multipliers of some de Branges-Rovnyak spaces...

We obtain a sufficient condition on a B(H)-valued function φ for the operator $⨍\mapsto {\Gamma }_{\phi }{⨍}^{\text{'}}\left(S\right)$ to be completely bounded on $H^{\infty }B\left(H\right)$; the Foiaş-Williams-Peller operator | St Γφ | Rφ = | | | 0 S | is then similar to a contraction. We show that if ⨍ : D → B(H) is a bounded analytic function for which $(1-r)\left|\right|{⨍}^{\text{'}}\left(r{e}^{i\theta }\right){\left|\right|}_{B\left(H\right)}^{2}rdrd\theta$ and $(1-r)\left|\right|⨍"\left(r{e}^{i\theta }\right){\left|\right|}_{B\left(H\right)}rdrd\theta$ are Carleson measures, then ⨍ multiplies ${\left(H^1{c}^1\right)}^{\text{'}}$ to itself. Such ⨍ form an algebra A, and when φ’∈ BMO(B(H)), the map $⨍\mapsto {\Gamma }_{\phi }{⨍}^{\text{'}}\left(S\right)$ is bounded $A\to B(H^2\left(H\right),L^2\left(H\right)\ominus H^2\left(H\right))$. Thus we construct a functional calculus for operators of Foiaş-Williams-Peller type.

For subspaces, $X$ and $Y$, of the space, $D$, of all derivatives $M(X,Y)$ denotes the set of all $g\in D$ such that $fg\in Y$ for all $f\in X$. Subspaces of $D$ are defined depending on a parameter $p\in [0,\infty ]$. In Section 6, $M(X,D)$ is determined for each of these subspaces and in Section 7, $M(X,Y)$ is found for $X$ and $Y$ any of these subspaces. In Section 3, $M(X,D)$ is determined for other spaces of functions on $[0,1]$ related to continuity and higher order differentiation.