Earl Berkson

Stečkin's Theorem, transference, and spectral decompositions

Abstract Let Y be a closed subspace of Lp(μ), where μ is an arbitrary measure and 1 sup {¦|V n ¦|: n = 0, ±1, ±2,…} (in particular, every surjective isometry of Y) can be expressed in the form V = eiA, where A is well bounded of type (B) (i.e., A has a spectral “diagonalization” analogous to, though weaker than, that in the spectral theorem for self-adjoint operators). This result, which fails if Y is replaced by an arbitrary reflexive space, is obtained by a blend of the transference method of Coifman and Weiss with Steckins Theorem and a recent result in abstract operator theory. It has the direct consequence that every uniformly bounded one-parameter group on Y is the Fourier-Stieltjes transform of a projection-valued mapping of R . An additional consequence is that every hermitian-equivalent operator on Y is well bounded of type (B). In the setting of an arbitrary Banach space X, power-bounded operators with a logarithm of the form iA with A well bounded of type (B) are studied. It is shown that if U is such an operator on X, then for every function f of bounded variation on the unit circle, ∑ n = − ∞ ∞ \ tf(n) U n converges in the strong operator topology. This result, which formally is a transference by U of Steckins Theorem, makes it possible to calculate directly from U a (normalized) logarithm for U and the spectral projections for the logarithm.

Spectral families of projections, semigroups, and differential operators

This paper presents new developments in abstract spectral theory suitable for treating classical differential and translation operators. The methods are specifically geared to conditional convergence such as arises in Fourier expansions and in Fourier inversion in general. The underlying notions are spectral family of projections and well-bounded operator, due to D. R. Smart and J. R. Ringrose. The theory of well-bounded operators is considerably expanded by the introduction of a class of operators with a suitable polar decomposition. These operators, called polar operators, have a canonical polar decomposition, are free from restrictions on their spectra (in contrast to well-bounded operators), and lend themselves to semigroup considerations. In particular, a generalization to arbitrary Banach spaces of Stones theorem for unitary groups is obtained. The functional calculus for well-bounded operators with spectra in a nonclosed arc is used to study closed, densely defined operators with a well-bounded resolvent. Such an operator L is represented as an integral with respect to the spectral family of its resolvent, and a sufficient condition is given for (-L) to generate a strongly continuous semigroup. This approach is applied to a large class of ordinary differential operators. It is shown that this class contains significant subclasses of operators which have a polar resolvent or generate strongly continuous semigroups. Some of the latter consist of polar operators up to perturbation by a semigroup continuous in the uniform operator topology.

TRANSFERENCE OF STRONG TYPE MAXIMAL INEQUALITIES BY SEPARATION-PRESERVING REPRESENTATIONS

Improved dry cleaning formulation containing a dry cleaning solvent, water, inorganic polyphosphate salt, hydrogen peroxide and a suitable detergent surfactant having a pH value of from 5 to 9, which minimizes equipment corrosion and maintains fabric strength while effectively removing hydrophilic stains.

Fourier series criteria for operator decomposability

Let U be an invertible operator on a Banach space Y. U is said to betrigonometricallywell-bounded provided the sequence {Un}n∞=−∞ is the Fourier-Stieltjes transform of a suitable projection-valued function E(·): [0, 2π]→ℬ(Y). This class of operators is known to apply naturally to a variety of classical phenomena which exclude the presence of spectral measures. In the case Y reflexive we use the Cesáro means σn(U, t) of the trigonometric series ∑k≠0 k−eiktUk, whichformally transfers the discrete Hilbert transform to Y, in order to give three separate necessary and sufficient conditions for U to be trigonometrically well-bounded. One of these conditions is sup {∥σn(U,t)∥: n ≥ 1, t ∈ [0,2π]} < ∞

Some types of Banach spaces, Hermitian operators, and Bade functionals

Introduction. In [6] and [11] a general notion of hermitian operator has been developed for arbitrary complex Banach spaces (see ?1 below). In terms of this notion, a family of operators on a Banach space is said to be hermitian-equivalent if the operators of this family can be made simultaneously hermitian by equivalent renorming of the underlying space [ 7]. Let X be a complex Banach space with norm 11 11, and let F be a commutative hermitian-equivalent family of operators on X. A norm for X equivalent to 11 11, and relative to which the operators of F are hermitian will be called an F-norm. Such families have been studied in [ 7], where it is shown that if X is a Hilbert space, then there is an F-norm which is also a Hilbert space norm. It is natural to seek other properties which, if enjoyed by X, can be preserved by choosing an F-norm appropriately. Such an investigation is conducted in this paper. Specifically, we show in ??3 and 4 that if the Banach space X has uniformly Frechet differentiable norm (resp., is uniformly convex), then there is an F-norm which preserves uniform Frechet differentiability (resp., uniform convexity). Moreover, we show in ?6 that if X = L4u), m > p > 1, u a measure, then there is an F-norm which preserves both uniform Frechet differentiability and uniform convexity. Our result for the case where X is uniformly convex enables us to establish in Theorem (5.4) a strong link between the notions of semi-innerproduct (see ?1) and Bade functional. This link adds to the analogy with Hilbert space inherent in these notions. Throughout this paper all spaces are over the complex field, and an operator will be a bounded linear transformation with range contained in its domain. In some cases it will be convenient to employ a notation for Banach spaces which explicitly exhibits the norm. Thus, if Y is a linear space, and I I is a Banach space norm for Y, we shall sometimes designate the resulting Banach space by (Y,l 1).

Multipliers for weighted LP-spaces, transference, and the q-variation of functions

Abstract If 1 f → (π f ) v , defined initially on the Schwartz class, extends to a bounded linear transformation of Lp(ω) into itself. For 1 ≤ q M q(R) denote the algebra of functions fϵL∞(R) such that the q-variation of f over the dyadic intervals is uniformly bounded. We show that if 2 ≤ p ≤ ∞ and ω ϵ Ap/2(R), then there is a real number s > 2 such that M q(R) ⊆ Mp,ω(R) for 1 ≤ q w ), w ϵ Ap/2(Z), by developing some machinery for transferring multipliers from one weighted setting to another. Our result on the inclusion M q(R) ⊆ Mp,ω(R) states that, under suitable circumstances, Kurtzs weighted Marcinkiewicz Multiplier Theorem extends from M 1(R) to M q(R), and also furnishes a counterpart for the (unweighted) generalization of the classical Marcinkiewicz Multiplier Theorem due to Coifman , de Francia , and Semmes .

Representations of groups with ordered duals and generalized analyticity

Let G be a locally compact abelian group whose dual group G has a Haar-measurable order. We show that for each strongly continuous, uniformly bounded representation R of G in a UMD space X, there is a corresponding direct-sum decomposition of X which reflects the order in G. The projections in X corresponding to this direct-sum decomposition have norms controlled solely by the bound of R and a constant depending only on X. We illustrate how this “vector-valued harmonic conjugation” result generalizes the various abstract successors of the M. Riesz theorem and we introduce an application to the superdiagonalization of kernels for abstract integral operators.

On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces

Let X be a closed subspace of LP(μ), where μ is an arbitrary measure and 1<p<∞. For an invertible power-bounded linear operator U: X→X and n=1,2,..., letA(n) and ℋ(n) denote the discrete ergodic averages and Hilbert transform truncates defined by U. We extend to this setting the μ-a. e. convergence criteria forA(n) and ℋ(n) which V. F. Gaposhkin and R. Jajte introduced for unitary operators on L2(μ). Our methods lift the setting from X to ℓp, where classical harmonic analysis and interpolation can be applied to suitable square functions.

A generalization of Macaev's theorem to non-commutative LP-spaces

For 1<p<∞ the non-commutative LP-spaces associated with a von Neumann algebra are shown to belong to the class UMD (that is, to possess the unconditionality property for martingale differences). With the aid of a recent result of the authors, which permits the classical Hilbert transform to be transferred to UMD spaces, a generalization of Macaevs theorem to non-commutative LP-spaces is introduced. This generalization utilizes the Hilbert kernel in a central role, broadens the “harmonic conjugation” aspects of Macaevs theorem, and provides a universal bound depending only on p.

Spectral families of projections in Hardy spaces

Strongly continuous one-parameter groups of isometries in the reflexive Hardy spaces of the disc D and the half-plane are considered in the light of the authors previous joint result with H. Benzinger and T. A. Gillespie which generalizes Stones theorem for unitary groups to arbitrary Banach spaces. It is shown that every such group {Tt} of Hardy space isometries has a spectral decomposition (with respect to a suitable projection-valued function on the real line R), as in the classical statement of Stones theorem in Hilbert space. (The relevant type of projection-valued function is known as a “spectral family.”) This circle of ideas is intimately bound up with harmonic analysis, particularly in HpR. In particular, if the group {Tt} acts in HpD and is associated as in Forellis theorem with a group of parabolic Mobius transformations of D, then it can be analyzed by way of the translation group on HpR. The Stone-type spectral family of the latter is shown to be obtained by restriction of the M. Riesz projections to HpR. By this means a concrete description of the Stone-type spectral family for a parabolic isometric group on HpD is obtained. A pleasant by-product of the parabolic case is absorption of the classical Paley-Wiener theorem for HpR, 1 < p ⩽ 2, into the framework of the generalized Stones theorem.