Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

We prove comparison theorems for the H ^∞-calculus that allow to transfer the property of having a bounded H ^∞-calculus from one sectorial operator to another. The basic technical ingredient are suitable square function estimates. These comparison results provide a new approach to perturbation theorems for the H ^∞-calculus in a variety of situations suitable for applications. Our square function estimates also give rise to a new interpolation method, the Rademacher interpolation. We show that a bounded H ^∞-calculus is characterized by interpolation of the domains of fractional powers with respect to Rademacher interpolation. This leads to comparison and perturbation results for operators defined in interpolation scales such as the L _p -scale. We apply the results to give new proofs on the H ^∞-calculus for elliptic differential operators, including Schrödinger operators and perturbed boundary conditions. As new results we prove that elliptic boundary value problems with bounded uniformly coefficients have a bounded H ^∞-calculus in certain Sobolev spaces and that the Stokes operator on bounded domains Ω with ?Ω ∈ C ^1,1 has a bounded H ^∞-calculus in the Helmholtz scale L _p,σ (Ω), p ∈ (1,∞).     

Analytic factorizations and completely bounded maps

We prove an analytic factorization theorem in the setting of the recently developed theory of operator spaces. We especially obtain the following result: LetA be aC ^*-algebra andH be a Hilbert space. Let π be an element ofH ^∞ (CB(A, B(H))), i.e. a bounded analytic function valued in the space of completely bounded maps fromA intoB(H). Then there exist a Hilbert spaceK, a representation π:A→B(K), ?₁ ∈₁ H ^∞ (B(H,K)) and ∈₂ H ^∞ (B(K,H)) such that ‖ε₁‖∞‖∈₂‖∞ ≤ ‖∈‖∞ and: $\forall z \in D, \forall a \in A, \varphi (z)(a) = \varphi _2 (z)\pi (a)\varphi _1 (z).$ We also prove an analogous result for completely bounded multilinear maps. The last part of the paper is devoted to a new proof of Pisier's theorem about gamma-norms.     

Algebras generated by inner functions

Let A be a uniform algebra on a compact space X. An inner function is a function in A unimodular on X. For three algebras of type H^∞ we prove A is generated by its inner functions. Whenever A is generated by its inner functions we prove the unit ball of A is the closed convex ball of the inner functions.     

Boundary values of differentiable functions defined on an arbitrary domain of a Carnot group

We study the boundary values of the functions of the Sobolev function spaces W _∞ ^l and the Nikol’ski? function spaces H _∞ ^l which are defined on an arbitrary domain of a Carnot group. We obtain some reversible characteristics of the traces of the spaces under consideration on the boundary of the domain of definition and sufficient conditions for extension of the functions of these spaces outside the domain of definition. In some cases these sufficient conditions are necessary.     

On well-posedness of the initial boundary-value problem for the derivative nonlinear Schrödinger equation

This paper deals with an initial boundary-value problem for the generalized derivative nonlinear Schrödinger equation. The cases of zero Dirichlet and generalized periodic boundary conditions are considered. The global existence of a solution inL _∞(0,∞;H _b ¹) is proved. The uniqueness inL _∞(0,T;H _b ¹)∩{u: ?u/?x εL _∞(Ω×(0,T))} is also established.     

Criteria for the nuclearity of spaces of functions of infinitely many variables

The present paper evolves from Berezanskii and Gali (Ukrainian Math. J.24 (4) (1972), 435–464) and Berezanskii, Gali, and Zuk (Soviet. Math. Dokl.13 (2) (1972)), in which, it was shown how one can construct a weighted infinite tensor product H_e,δ = ?_{n = 1:e,δ}^∞ = H_τn of Hilbert spaces H_τn with a given stabilizing sequence δ = (δ_n)_{n = 1}^∞(δ_n > 0). Here a weighted infinite tensor product ?_e = ?_{n = 1,e}^∞?_n of nuclear spaces ?_n is established first. Criteria for nuclearity of the constructed spaces are also given. Some examples of nuclear spaces of functions of infinite many variables K(T^∞) and A(T^∞) are obtained.     

Dual spaces of C∞ (X)

ForX a locally compact Stonian Space, letC _∞ (X) denote the universally complete Riesz space of all extended-real-valued continuous functionsf onX for which {x∈X| |f (x)|=∞} is nowhere dense. In this paper the dual spaces ofC _∞ (X) (i.e. the spaces of order bounded; of σ-order continuous; of order continuous linear forms onC _∞ (X), and the extended order dual ofC _∞ (X) denote here byC _∞ (X)^ρ (introduced by W.A.J. Luxemburg and J.J. Masterson)) are characterized. It is shown thatC _∞ (X)^ρ can be identified in a canonical way with the inductive limitM _q (X) of the Riesz spaces of all normal Radon measures defined on the dense open subsets ofX. More generally, ifY is a locally compact space thenM _q (Y) is the extended order dual of the inductive limit of the Riesz spaces of all real-valued continuous functions defined on the dense open subsets ofY. IfX is locally compact and hyperstonian, then it is proved thatC _∞ (X) andC _∞ (X)^ρ are isomorphic, and a criterion forC _∞ (X)^ρ to be the universal completion of the space of order continuous linear forms onC _∞ (X) is given.     

Coherence of some rings of functions

To say that a commutative ring $\text{R}$ with unit is coherent amounts to saying, in case $\text{R}$ has no divisors of zero, that the intersection of two finitely generated ideals in $\text{R}$ is finitely generated. We prove that the ring H^∞ of bounded analytic functions in the unit disc is coherent, while the disc algebra A is not coherent. For any positive measure μ, L^∞(μ) is coherent.     

Topological stable rank of H ∞(Ω) for circular domains Ω

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ^∞(Ω) the Banach algebra of all bounded holomorphic functions on Ω, with pointwise operations and the supremum norm. We show that the topological stable rank of H ^∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topological stable rank of H ^∞( $ \mathbb{D} $ ) is equal to 2, where $ \mathbb{D} $ is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H _? ^∞ (Ω) are 2.     

Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane

It is proved that there exists a function defined in the closed upper half-plane for which the sums of its real shifts are dense in all Hardy spaces H_p for 2 ≤ p < ∞, as well as in the space of functions analytic in the upper half-plane, continuous on its closure, and tending to zero at infinity.

     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Estimates in the Carleson corona theorem,ideals of the algebra H∞, a problem of S.-Nagy

Let E₁, E₂, be Hilbert spaces, H^∞(E₁,E₂) be the space of functions, bounded and analytic in the disk D, with values in the space of bounded linear operators from E₁ to E₂. Estimates are investigated for a solution of the problem of S.-Nagy of finding a left inverse element for a function F, FεH^∞(E₁,E₂). For dim E₁=1 this problem is a generalization of the corona problem. Let C_n(δ)= sup_¦G¦_∞∶FεH^∞(E₁,E₂),dim E₁=n, ¦F¦_∞?1, ¦F(z)a¦₂?δ¦a¦₂(zεD,aεE₁);Gε H^∞(E₂,E₁) is a function of minimal norm for which . Then where a_n, C_n are constants depending only on n. The behavior of the function C₁ as δ→1 is described. Other results are obtained also.     

Rickart C1 algebras,II

The theory of inner-outer factorization in the Hardy spaces H^p in the unit disc $\text{D}$ is well known and has many applications. It does not carry over to the spaces H^p on the polydisc $\text{D}$ⁿ or the ball $\text{B}$ⁿ when n > 1. However, for Lumer's Hardy spaces (LH)^p on any simply connected complex analytic manifold, we introduce the notions of internal and external functions and prove that every f? (LH)^p has a factorization f = I_ε × E_ε, where I_ε is internal and E_ε is external, and E_ε? (LH)^p?ε, for any ε > 0. The factorization is not unique and an example of Rudin shows that the ε is needed, at least when $\text{p =}\text{2}\text{m}$, where m is an integer.     

The L ∞-Stokes semigroup in exterior domains

The Stokes semigroup on a bounded domain is an analytic semigroup on spaces of bounded functions as was recently shown by the authors based on an a priori L ^∞-estimate for solutions to the linear Stokes equations. In this paper, we extend our approach to exterior domains and prove that the Stokes semigroup is uniquely extendable to an analytic semigroup on spaces of bounded functions.     

Inner functions in the unit ball of Cn

Let B be the open unit ball of $\text{C}$ⁿ, n > 1. Let I (for “inner”) be the set of all u ? H °(B) that have $\text{¦u¦ = 1}$ a.e. on the boundary S of B. Aleksandrov proved recently that there exist nonconstant u ? I. This paper strengthens his basic theorem and provides further information about I and the algebra Q generated by I. Let XY be the finite linear span of products xy, x ? X, y ? Y, and let $\text{¦X¦}$ be the norm closure, in L^∞ = L^∞(S), of X. Some results: set I is dense in the unit ball of H^∞(B) in the compact-open topology. On $\text{S,}\text{Q}\text{?}\text{Q}$ is weak¹-dense in $\text{L}{}^{\mathrm{\infty }}\text{, ¦}\text{Q}\text{?}\text{Q¦}$ does not contain $\text{H}{}^{\mathrm{\infty }}\text{, C(S) ?¦}\text{Q}\text{?}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ ¦}\text{H}\text{?}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}\text{¦ ≠ L}{}^{\mathrm{\infty }}$. (When $\text{n = 1, ¦Q¦ = H}{}^{\mathrm{\infty }}\text{and}\text{¦}\text{Q}\text{?}\text{Q¦ = L}{}^{\mathrm{\infty }}$.) Every unimodular $\text{?}\text{?}\text{L}{}^{\mathrm{\infty }}$ is a pointwise limit a.e. of products $\text{u}\text{v}\text{?}\text{, u}\text{?}\text{I, ν}\text{?}\text{I}$. The zeros of every $\text{? ? 0}$ in the ball algebra (but not of every H^∞-function) can be matched by those of some u ? I, as can any finite number of derivatives at 0 if $\text{∥?∥}{}_{\mathrm{\infty }}\text{< 1}$. However, $\text{?}\text{u}$ cannot be bounded in B if u ? I is non-constant.     

Invariant means and invariant ideals in L∞(G) for a locally compact group G

For a nondiscrete σ-compact locally compact Hausdorff group G, L_∞(G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals {N_α} such that N_α1 + N_α2 = L_∞(G) whenever α₁ ≠ α₂. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM?TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L_∞(G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.     

Norm-attaining weighted composition operators on weighted Banach spaces of analytic functions

We investigate weighted composition operators that attain their norm on weighted Banach spaces of holomorphic functions on the unit disc of type H ^∞. Applications for composition operators on weighted Bloch spaces are given.     

Paley-Wiener-Type Theorems for a Class of Integral Transforms

A characterization of weighted L₂(I) spaces in terms of their images under various integral transformations is derived, where I is an interval (finite or infinite). This characterization is then used to derive Paley-Wiener-type theorems for these spaces. Unlike the classical Paley-Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane. The class of integral transformations considered is related to singular Sturm-Liouville boundary-value problems on a half line and on the whole line.