Richard J. Fleming

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 Schur D-stable matrices

Abstract It is shown that vertex stability implies Schur D-stability for real 2 × 2 matrices and real n × n tridiagonal matrices. Additional results describing the class of n × n complex Schur D-stable matrices are given.

Extreme Point Methods and Banach-Stone Theorems

An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0.Q; X/ into C0.K; Y/ require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.

Classes of Schur D-stable matrices

Abstract It is shown that vertex stability implies Schur D-stability for real 3×3 matrices. Also, principally nilpotent n×n complex matrices are shown to be perfectly Schur D-stable, and additional characterizations of these matrices are given.

Normal and quasinormal weighted composition operators

In their paper [1], Campbell and Jamison attempted to give necessary and sufficient conditions for a weighted composition operator on an L 2 space to be normal, and to be quasinormal. Those conditions, specifically Theorems I and II of that paper, are not valid (see [2] for precise comments on the other results in that paper). In this paper we present a counterexample to those theorems and state and prove characterizations of quasinormality (Theorem 1 below) and normality (Theorem 2 and Corollary 3 below). We also discuss additional examples and information concerning normal weighted composition operators which contribute to the further understanding of this class.

Extreme contractions on continuous vector-valued function spaces

An old question asks whether extreme contractions on C(K) are necessarily nice; that is, whether the conjugate of such an operator maps extreme points of the dual ball to extreme points. Partial results have been obtained. Determining which operators are extreme seems to be a difficult task, even in the scalar case. Here we consider the case of extreme contractions on C(K,E), where E itself is a Banach space. We show that every extreme contraction T on C(K, E) to itself which maps extreme points to elements of norm one is nice, where K is compact and E is the sequence space c 0 .

Commutative ranges of analytic functions in Banach algebras

Let A denote a complex unital Banach algebra with Hermitian elements H( A). We show that if F is an analytic function from a connected open set D into A such that F(z) is normal (F(z) = u(z) + iv(z), where u(z), v(z) E H(A) and u(z)v(z) = v(z)u(z)) for each z E D, then F(z)F(w) = F(w)F(z) for all w, z E D. This generalizes a theorem of Globevnik and Vidav concerning operator-valued analytic functions. As a corollary, it follows that an essentially normal-valued analytic function has an essentially commutative range. In this note we extend to the Banach algebra setting a theorem of Globevnik and Vidav [7] which states that the set of values of a normal-operator-valued function, defined and analytic on an open connected set in the complex plane, is commutative. Suppose A is a complex unital Banach algebra with Hermitian elements H(A). Recall that an element u E A is said to be Hermitian if its numerical range V(u) = {f(u): f(1) = 1, If II = 1,f E A*} lies on the real axis. An element w E A is said to be normal if w = u + iv, where u, v E H(A) and uv = vu. We let J(A) = H(A) + iH(A). Then J(A) is a Banach subspace of A (but not necessarily a subalgebra) which contains both the Hermitian and normal elements. (See [4] for a discussion of these notions.) The mapping * from J(A) to itself, defined by (u + iv)* = u iv, is a continuous linear involution on J(A). Furthermore, an element w of J(A) is normal if and only if w*w = ww*. Suppose that F: D -+ A, where D is the open unit disk, is analytic and normal valued for each z E D. Thus F(z) = E=0u zi Ee J(A) for each z E D, where uj E A for eachj. THEOREM. Let A be a unital Banach algebra. Let F be analytic on the open unit disk D and let F( z) E A be normal for all z E D. Then there exist commuting normal elements u; e A satisfying

Some Banach Spaces on Which All Biholomorphic Automorphisms Are Linear

The question of whether biholomorphic maps are linear has been treated in various forms by several authors. In particular, Harris [I ] has shown that a biholomorphic map of the unit ball of one space to another which takes 0 to 0 is a restriction of a linear isometry between the two spaces. He then showed that if the unit ball of a Banach space is a homogeneous domain, then it is holomorphically equivalent to the unit ball of another Banach space if and only if the two spaces are isometrically isomorphic. He asked whether this result would hold without the assumption about a homogeneous domain. Kaup and Upmeier [2] gave an answer to this question by showing that two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent. In a recent paper. Stacho [5 1 gave a short proof of the fact that all biholomorphic automorphisms of the unit ball in certain Lp-spaces are linear. In the present note, we show how Stacho’s method can be used to obtain the same result for the C,-spaces and the spaces LP(R. E), where E is an arbitrary Banach space. R is a-finite, and 1 <p < + co, p # 2. In particular. for the discrete case, we get the result that I,(E) has the linear biholomorphic property whether E has the property or not. On the other hand, we show that c,(E) has the property if and only if E has it. A function Q on the open unit ball B(E) of a Banach space is said to be holomorphic in B(E) if the Frechet derivative D@(x, .) of 4 at x exists as a bounded linear map of E into E for each x E B(E). A function 4 from B(E) to B(E) is biholomorphic if 4-l exists and both + and # ’ are holomorphic. The proofs of Theorems 1 and 2 below as well as the theorem in IS] are based on the following lemma proved by Stacho in 1-S 1.

A note on operator norm inequalities

If P is a positive operator on a Hilbert space H whose range is dense, then a theorem of Foias, Ong, and Rosenthal says that: ‖[ϕ(P)]−1T[ϕ(P)]‖<-12 max {‖T‖, ‖P−1TP‖} for any bounded operator T on H, where φ is a continuous, concave, nonnegative, nondecreasing function on [0, ‖P‖]. This inequality is extended to the class of normal operators with dense range to obtain the inequality ‖[φ(N)]−1T[φ(N)]‖<-12c2 max {tT‖, ‖N−1TN‖} where φ is a complex valued function in a class of functions called vase-like, and c is a constant which is associated with φ by the definition of vase-like. As a corollary, it is shown that the reflexive lattice of operator ranges generated by the range NH of a normal operator N consists of the ranges of all operators of the form φ(N), where φ is vase-like. Similar results are obtained for scalar-type spectral operators on a Hilbert space.