The Neumann problem for some degenerate elliptic equations

In the paper we study the equation L _u = f, where L is a degenerate elliptic operator, with Neumann boundary condition in a bounded open set μ. We prove existence and uniqueness of solutions in the space H(μ) for the Neumann problem.     

Über Maximale Monotonie von Operatoren des Typs L*ϕ·L

In a Hilbert space H, we consider operators of type A=L^*ϕ·L, where L is a closed, linear operator and ϕ is a maximal cyclically monotone, coercive operator. The operators ϕ, L, L^* and their inverses are not necessarily everywhere defined. Our principle result is a nonlinear extension of an earlier theorem of v. Neumann for A=L^*L.Theorem: Suppose that either (L^*)⁻¹ is bounded or that both L⁻¹ is bounded and, D(ϕ) υ N (L^*). The L^*ϕ·L, is maximal cyclically monotone. Maximality of sums is also considered, and the theory is applied to concrete differential operators of the form , with monotone functions f₁ and various boundary conditions.      

Grothendieck’s theorem for operator spaces

We prove several versions of Grothendieck’s Theorem for completely bounded linear maps T:E→F ^*, when E and F are operator spaces. We prove that if E, F are C ^*-algebras, of which at least one is exact, then every completely bounded T:E→F ^* can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as T=T _r +T _c where T _r (resp. T _c ) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on C ^*-algebras. Moreover, our result holds more generally for any pair E, F of “exact” operator spaces. This yields a characterization of the completely bounded maps from a C ^*-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual E ^* are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class. Oblatum 31-I-2002 & 3-IV-2002?Published online: 17 June 2002     

Nonself-Adjoint Operators with Almost Hermitian Spectrum: Weak Annihilators

We consider nonself-adjoint nondissipative trace class additive perturbations L=A+iV of a bounded self-adjoint operator A in a Hilbert space ,H. The main goal is to study the properties of the singular spectral subspace N _i ⁰ of L corresponding to part of the real singular spectrum and playing a special role in spectral theory of nonself-adjoint nondissipative operators.To some extent, the properties of N _i ⁰ resemble those of the singular spectral subspace of a self-adjoint operator. Namely, we prove that L and the adjoint operator ,L ^* are weakly annihilated by some scalar bounded outer analytic functions if and only if both of them satisfy the condition N _i ⁰ =H. This is a generalization of the well-known Cayley identity to nonself-adjoint operators of the above-mentioned class.     

Feedback operator for a pseudoparabolic optimal control problem

The feedback operator of a linear pseudoparabolic problem with quadratic criterion is obtained by decoupling of the optimality condition. The feedback operator is shown to be related to the solution of a Riccati equation formulated in theB*-algebra of bounded linear operators onL ²(). This approach shows that the linear feedback operator may be considered as a bounded operator fromL ²() intoH ₀ ¹ (). Finally, we give a theorem establishing the convergence behavior for the feedback operators for these problems as they formally approach an analogous problem of parabolic type.This work was supported in part by the National Science Foundation, Grant No. MCS-7902037.     

The conjugate gradient method for linear ill-posed problems with operator perturbations

We consider an ill-posed problem Ta = f_* in Hilbert spaces and suppose that the linear bounded operator T is approximately available, with a known estimate for the operator perturbation at the solution. As a numerical scheme the CGNR-method is considered, that is, the classical method of conjugate gradients by Hestenes and Stiefel applied to the associated normal equations. Two a posteriori stopping rules are introduced, and convergence results are provided for the corresponding approximations, respectively. As a specific application, a parameter estimation problem is considered.     

The weak* similarity property on dual operator algebras

This paper is devoted to dual operator algebras, that isw ^*-closed algebras of bounded operators on Hilbert space. We investigate unital dual operator algebrasA with the following weak^* similarity property: for every Hilbert spaceH, anyw ^*-continuous unital homomorphism fromA intoB(H) is completely bounded and thus similar to a contractive one. We develop a notion of dual similarity degree for these algebras, in analogy with some recent work of Pisier on the similarity problem on operator algebras.     

On the square root of a positive B( )-valued function

Let (Ω, , μ) be a measure space, a separable Banach space, and ^* the space of all bounded conjugate linear functionals on . Let f be a weak^* summable positive B( ^*)-valued function defined on Ω. The existence of a separable Hilbert space , a weakly measurable B( )-valued function Q satisfying the relation Q^*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L_2,f of square integrable operator-valued functions with respect to f. It is shown that for B⁺( ^*)-valued measures, the concepts of weak^*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained.     

Walsh Multipliers for Dyadic Hardy Spaces

In this article we are interested in conditions on the coefficients of a Walsh multiplier operator that imply the operator is bounded on certain dyadic Hardy spaces H ^p , 0 < p < ∞. In particular, we consider two classical coefficient conditions, originally introduced for the trigonometric case, the Marcinkiewicz and the Hörmander–Mihlin conditions. They are known to be sufficient in the spaces L ^p , 1 < p < ∞. Here we study the corresponding problem on dyadic Hardy spaces, and find the values of p for which these conditions are sufficient. Then, we consider the cases of H ¹ and L ¹ which are of special interest. Finally, based on a recent integrability condition for Walsh series, a new condition is provided that implies that the multiplier operator is bounded from L ¹ to L ¹, and from H ¹ to H ¹. We note that existing multiplier theorems for Hardy spaces give growth conditions on the dyadic blocks of the Walsh series of the kernel, but these growth are not computable directly in terms of the coefficients.     

Topological Structure of the Set of Weighted Composition Operators on Weighted Bergman Spaces of Infinite Order

We consider the topological space of all weighted composition operators on weighted Bergman spaces of infinite order endowed with the operator norm. We show that the set of compact weighted composition operators is path connected. Furthermore, we find conditions to ensure that two weighted composition operators are in the same path connected component if the difference of them is compact. Moreover, we compare the topologies induced by L(H^∞) and L(H^∞_v) on the space of bounded composition operators and give a sufficient condition for a composition operator to be isolated.     

On the differentiability of very weak solutions with right-hand side data integrable with respect to the distance to the boundary

We study the differentiability of very weak solutions v∈L¹(Ω) of ₀(v,L^?φ)=₀(f,φ) for all vanishing at the boundary whenever f is in L¹(Ω,δ), with δ=dist(x,∂Ω), and L^* is a linear second order elliptic operator with variable coefficients. We show that our results are optimal. We use symmetrization techniques to derive the regularity in Lorentz spaces or to consider the radial solution associated to the increasing radial rearrangement function of f.     

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

Bilinear forms on exact operator spaces andB(H)⊗B(H)

LetE, F be exact operator spaces (for example subspaces of theC ^*-algebraK(H) of all the compact operators on an infinite dimensional Hilbert spaceH). We study a class of bounded linear mapsu: E →F ^* which we call tracially bounded. In particular, we prove that every completely bounded (in shortc.b.) mapu: E →F ^* factors boundedly through a Hilbert space. This is used to show that the setOS _n of alln-dimensional operator spaces equipped with thec.b. version of the Banach Mazur distance is not separable ifn>2. As an application we whow that there is more than oneC ^*-norm onB (H) ? B (H), or equivalently that $$B(H) \otimes _{\min } B(H) \ne B(H) \otimes _{\max } B(H),$$ which answers a long standing open question. Finally we show that every “maximal” operator space (in the sense of Blecher-Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the “exactness constant”. In the final section, we introduce and study a new tensor product forC ^*-albegras and for operator spaces, closely related to the preceding results.     

Schatten class hankel operators on the Bergman space

In this paper we characterize Hankel operatorsH _f andH _f on the Bergman spaces of bounded symmetric domains which are in the Schatten p-class for 2p< and f inL ² using a Jordan algebra characterization of bounded symmetric domains and properties of the Bergman metric.     

Least-squares solutions of multi-valued linear operator equations in Hilbert spaces

Let M be a linear manifold in H₁ H₂, where H₁, and H₂ are Hilbert spaces. Two notions of least-squares solutions for the multi-valued linear operator equation (inclusion) y ε M(x) are introduced and investigated. The main results include (i) equivalent conditions for least-squares solvability, (ii) properties of a least-squares solution, (iii) characterizations of the set of all least-squares solutions in terms of algebraic operator parts and generalized inverses of linear manifolds, and (iv) best approximation properties of generalized inverses and operator parts of multi-valued linear operators. The principal tools in this investigation are an abstract adjoint theory, orthogonal operator parts, and orthogonal generalized inverses of linear manifolds in Hilbert spaces.     

Ergodic theory and iterative solution of linear operator equations

In this paper we consider the problem of approximating solutions of linear operator equations of the type u-Tu=f. The main tools are Dotson's extension of the Eberlein ergodic theorem to affine mappings and the DeMoivre-Laplace theorem of probability theory. The main results are applied to obtain theorems on the iterative approximation of solutions of linear operator equations in Hilbert space and the approximation in L ^ρ norm of solutions of a certain functional equation in the space L ^∞     

The weak type inequality for the maximal operator of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series

The main aim of this paper is to prove that the maximal operator σ^* of the Marcinkiewicz–Fejér means of the two-dimensional Walsh–Fourier series is bounded from the Hardy space H_2/3 to the space weak-L_2/3.     

Fejér summability of multi-parameter bounded Ciesielski systems

We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space H _p([0,1)^d ₁ × ¨ × [0,1)^d _l) to L _p([0,1)^d) if 1/2 < p < &infin; and m _j &ge; 0, |k _j| &le; m _j + 1. By an interpolation theorem, we get that the maximal operator is also of weak type (H ₁ ^#i,L ₁) (I=1,¨,l), where the Hardy space H ₁ ^#i is defined by a hybrid maximal function and H ₁ ^#i L(logL)^l-1. As a consequence, we obtain that the Fejér means of the Ciesielski-Fourier series of a function f converge to f a.e. if f H ₁ ^#i and converge in a cone if f &isin; L ₁.     

Weighted inequalities and a.e. convergence for Poisson integrals in light-cones

We show that the Poisson maximal operator for the tube over the light-cone, P ^*, is bounded in the weighted space L ^p (w) if and only if the weight w(x) belongs to the Muckenhoupt class A _p . We also characterize with a geometric condition related to the intrinsic geometry of the cone the weights v(x) for which P ^* is bounded from L ^p (v) into L ^p (u), for some other weight u(x) > 0. Some applications to a.e. restricted convergence of Poisson integrals are given.     

Existence for a time‐dependent heat equation with non‐local radiation terms

The paper deals with the time‐dependent linear heat equation with a non‐linear and non‐local boundary condition that arises when considering the radiation balance. Solutions are considered to be functions with values in V := {v ∈ H¹(Ω)∣γv ∈ L₅(∂Ω)}. As a consequence one has to work with non‐standard Sobolev spaces. The existence of solutions was proved by using a Galerkin‐based approximation scheme. Because of the non‐Hilbert character of the space V and the non‐local character of the boundary conditions, convergence of the Galerkin approximations is difficult to prove. The advantage of this approach is that we don't have to make assumptions about sub‐ and supersolutions. Finally, continuity of the solutions with respect to time is analysed. Copyright © 1999 John Wiley & Sons, Ltd.