James Jamison

Isometries on Banach spaces : function spaces

BEGINNINGS Introduction Banachs Characterization of Isometries on C(Q) The Mazur-Ulam Theorem Orthogonality The Wold Decomposition Notes and Remarks CONTINUOUS FUNCTION SPACES--THE BANACK-STONE THEOREM Introduction Eilenbergs Theorem The Nonsurjective case A Theorem of Vesentini Notes and Remarks THE L(p) SPACES Introduction Lampertis Results Subspaces of L(p) and the Extension Theorem Bochner Kernels Notes and Remarks ISOMETRIES OF SPACES OF ANALYTIC FUNCTIONS Introduction Isometries of the Hardy Spaces of the disk Bergman spaces Bloch Spaces S(p) Spaces Notes and Remarks REARRANGEMENT INVARIANT SPACES Introduction Lumers Method for Orlicz Spaces Zaidenbergs Generalization Musielak-Orlicz Spaces Notes and Remarks BANACH ALGEBRAS Introduction Kadisons Theorem Subdifferentiability and Kadisons Theorem The Nonsurjective Case of Kadisons theorem The Algebras C(1) and AC Douglas Algebras Notes and Remarks BIBLIOGRAPHY INDEX

On some classes of weighted composition operators

Known regularity conditions (given below) are both necessary and sufficient for the linear transformation/—* WTf to define a bounded operator on L ((i); such operators are called weighted composition operators. In case w(x) = 1 the operator CT defined via composition with T is simply a composition operator. In this paper we characterize the normal, quasi-normal and hermitian weighted composition operators in terms of w, T, and dn°T~/dfi. Some known results on seminormal composition operators and weighted composition operators are obtained as corollaries. We given an example of quasinormal WT with non-constant weight which is not normal. Campbell and Dibrell [2] give sufficient conditions for a composition operator CT to be power hyponormal; that is, for (CT) n to be hyponormal for all natural numbers n. We give a sufficient condition for WT to be power hyponormal, generalizing in a natural way the corresponding result for weighted shifts on the integers.

Isometries of some Banach spaces of analytic functions

We characterize the surjective isometries of a class of analytic functions on the disk which include the Analytic Besov spaceBp and the Dirichlet spaceDp. In the case ofBp we are able to determine the form of all linear isometries on this space. The isometries for these spaces are finite rank perturbations of integral operators. This is in contrast with the classical results for the Hardy and Bergman spaces where the isometries are represented as weighted compositions induced by inner functions or automorphisms of the disk.

Shift Operators on Banach Spaces

Abstract J. R. Holub has obtained several results for shift operators on C ( X ). In this paper we answer some questions of Holub and obtain extensions of many of his results. In particular, we show that C ( X , R ) does not admit a shift operator if X has only countably many components and each component is infinite. We show that C ( X , C ) does not admit a shift operator for certain special compact Hausdorff spaces X . We show that there exists a compact Hausdorff space X which is not totally disconnected and both C ( X , C ) and C ( X , R ) admit shift operators. If 1 ⩽ p X , Σ , μ ) is a σ-finite non-atomic measure space then L P R ( μ ) does not admit a disjointness preserving shift operator. We also show that l P for 1 ⩽ p ⩽ ∞ is the only L P R ( μ ) space which admits a disjointness preserving shift operator.

Almost transitivity of some function spaces

The almost transitive norm problem is studied for L p (/i,X), C(K,X) and for certain Orlicz and Musielak—Orlicz spaces. For example if p =f= 2 <oo then L p (/i) has almost transitive norm if and only if the measure/* is homogeneous. It is shown that the only Musielak—Orlicz space with almost transitive norm is the L^-space. Furthermore, an Orlicz space has an almost transitive norm if and only if the norm is maximal. L p (/i,X) has almost transitive norm if L V (JJL) and X have. Separable spaces with nontrivial L p -structure fail to have transitive norms. Spaces with nontrivial centralizers

A learning rule with generalized Hebbian synapses

Abstract We study the convergence behavior of a learning model with generalized Hebbian synapses.

One-complemented subspaces of Musielak--Orlicz sequence spaces

The aim of this paper is to characterize one-complemented subspaces of finite codimension in the Musielak-Orlicz sequence space l Φ . We generalize the well-known fact (Ann. Mat. Pura Appl. 152 (1988) 53; Period. Math. Hungar. 22 (1991) 161; Classical Banach Spaces I, Springer, Berlin, 1977) that a subspace of finite codimension in l p , 1 ≤ p n ) we prove a similar characterization in l Φ . In the case of Orlicz spaces we obtain a complete characterization of one-complemented subspaces of finite codimension, which extends and completes the results in Randrianantoanina (Results Math. 33(1-2) (1998) 139). Further, we show that the well-known fact that a one-complemented subspace of finite codimension in l p , 1 ≤ p p -spaces, 1 2 -spaces, in terms of one-complemented hyperplanes, in the class of Musielak-Orlicz and Orlicz spaces as well.

Isometrically Equivalent Composition Operators on Spaces of Analytic Vector-Valued Functions

Let

Homomorphisms on a class of commutative Banach algebras

X

The Analysis of Composition Operators on LP and the Hopf Decomposition

be a Banach space and let