Recurrent real hypersurfaces in complex two-plane Grassmannians

Summary We introduce the notion of recurrent shape operator for a real hypersurface M in the complex two-plane Grassmannians G₂(C^m+2) and give a non-existence property of real hypersurfaces in G₂(C^m+2) with the recurrent shape operator.     

Some Open Problems and Conjectures Associated with the Invariant Subspace Problem

There is a subtle difference as far as the invariant subspace problem is concerned for operators acting on real Banach spaces and operators acting on complex Banach spaces. For instance, the classical hyperinvariant subspace theorem of Lomonosov [Funktsional. Anal. nal. i Prilozhen 7(3)(1973), 55–56. (Russian)], while true for complex Banach spaces is false for real Banach spaces. When one starts with a bounded operator on a real Banach space and then considers some “complexification technique” to extend the operator to a complex Banach space, there seems to be no pattern that indicates any connection between the invariant subspaces of the “real” operator and those of its “complexifications.” The purpose of this note is to examine two complexification methods of an operator T acting on a real Banach space and present some questions regarding the invariant subspaces of T and those of its complexifications Mathematics Subject Classification 1991: 47A15, 47C05, 47L20, 46B99 Y.A. Abramovich: 1945–2003 The research of Aliprantis is supported by the NSF Grants EIA-0075506, SES-0128039 and DMI-0122214 and the DOD Grant ACI-0325846     

REAL AND COMPLEX OPERATOR IDEALS

Abstract

The powerful concept of an operator ideal on the class of all Banach spaces makes sense in the real and in the complex case. In both settings we may, for example, consider compact, nuclear, or 2-summing operators, where the definitions are adapted to each other in a natural way. This paper deals with the question whether or not that fact is based on a general philosophy. Does there exists a one-to-one correspondence between “real properties” and “complex properties” defining an operator ideal? In other words, does there exist for every real operator ideal a uniquely determined corresponding complex ideal and vice versa?

Unfortunately, we are not abel to give a final answer. Nevertheless, some preliminary results are obtained. In particular, we construct for every real operator ideal a corresponding complex operator ideal and for every complex operator ideal a corresponding real one. However, we conjecture that there exists a complex operator ideal which can not be obtained from a real one by this construction.

The following approach is based on the observation that every complex Banach space can be viewed as a real Banach space with an isometry acting on it like the scalar multiplication by the imaginary unit i.     

Real Hypersurfaces in Complex Two-Plane Grassmannians

 The complex two-plane Grassmannian G ₂(C ^m+2 in equipped with both a K?hler and a quaternionic K?hler structure. By applying these two structures to the normal bundle of a real hypersurface M in G ₂(C ^m+2 one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G ₂ C ^m+2 , m≥3, for which these two distributions are invariant under the shape operator of M. Received 13 November 1996; in revised form 3 March 1997     

Characterizations of Arveson's Hardy Space

A Hardy-type space H ² _d in the unit ball B_d of C^d , which was recently introduced by Arveson [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], is appropriate for the operator theory of d-contractions. In this article, it is proved that H ² _d actually coincides with a Hardy-Sobolev space. This yields almost immediately some of the related results obtained in [W. Arveson (1998). Subalgebras of C*-algebras III: multivariable operator theory. Acta Math., 181, 159-228.], including the facts that H ² _d is not associated with any measure on C ^d ; and that the corresponding algebra of multipliers M ? H ^∞(B_d ) and the inclusion is proper. Finally, a function-theoretic version of von Neumann's inequality for the d-contractions is presented.     

Solution of the spectral problem for the curl and Stokes operators with periodic boundary conditions

In this paper, we establish relations between eigenvalues and eigenfunctions of the curl operator and Stokes operator (with periodic boundary conditions). These relations show that the curl operator is the square root of the Stokes operator with ν = 1. The multiplicity of the zero eigenvalue of the curl operator is infinite. The space L ₂(Q, 2π) is decomposed into a direct sum of eigenspaces of the operator curl. For any complex number λ, the equation rot u + λu = f and the Stokes equation −ν(Δv + λ ²v) + ∇p = f, div v = 0, are solved. Bibliography: 15 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 246–276.     

Local existence in time of solutions to the elliptic-hyperbolic Davey-Stewartson system without smallness condition on the data

We study the initial value problem for the elliptic-hyperbolic Davey-Stewartson systems {ie133-01} where {ie133-2},u is a complex valued function and φ is a real valued function. When (c ₁,c ₂) = (-1, 2) the system (*) is called DSI equation in the inverse scattering literature. Our purpose in this paper is to prove the local existence of a unique solution to (*) in the Sobolev spaceH ²(R ²) without the smallness condition on the data which were assumed in previous works [7], [17], [19], [26], Our result is a positive answer to Question 7 in [24].     

Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces

We consider Hausdorff operators generated by a function ϕ integrable in Lebesgue"s sense on either R or R ², and acting on the real Hardy space H ¹(R), or the product Hardy space H ¹¹(R×R), or one of the hybrid Hardy spaces H ¹⁰(R ²) and H ⁰¹(R ²), respectively. We give a necessary and sufficient condition in terms of ϕ that the Hausdorff operator generated by it commutes with the corresponding Hilbert transform. This revised version was published online in June 2006 with corrections to the Cover Date.     

Concave and positive summing norms for operators on Lp

Let p be a real number such that p ? [1,+ ￥] p \in [1,+ \infty] and its conjugate exponent p' is not an even integer and let T be an operator defined on L^p(l)L^p(\lambda ) with values in a Banach space. We prove that the image of the unit ball determines if T belongs to the space of concave and positive summing operators. We also prove that the image of the unit ball determines the representability of the operator.     

Thermodynamic Limits for Hamiltonians Defined as Pseudodifferential Operators

Abstract

If P(h) is a h-pseudodifferential operator in R ⁿ associated to an holomorphic semi-bounded symbol in some neighborhood of the real phase space, with bounded derivatives, we describe the symbol of e ^?tP(h), by inequalities where the constants depend on the bounds for the derivatives of the symbol of P(h), but not on the dimension n. Some applications to thermodynamic limits (free energy) are given.     

A characterization of totally <Emphasis Type="Italic">η</Emphasis>-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form

We give a characterization of totally η-umbilical real hypersurfaces and ruled real hypersurfaces of a complex space form in terms of totally umbilical condition for the holomorphic distribution on real hypersurfaces. We prove that if the shape operator A of a real hypersurface M of a complex space form M ⁿ (c), c ≠ 0, n ⩾ 3, satisfies g(AX, Y) = ag(X, Y) for any X, Y ∈ T ₀(x), a being a function, where T ₀ is the holomorphic distribution on M, then M is a totally η-umbilical real hypersurface or locally congruent to a ruled real hypersurface. This condition for the shape operator is a generalization of the notion of η-umbilical real hypersurfaces.     

Vafa-Witten bound on the complex projective space

We prove that the largest first eigenvalue of the Dirac operator among all Hermitian metrics on the complex projective space of odd dimension m, larger than the Fubini-Study metric is bounded by (2m(m+1))^1/2. Mathematics Subject Classification (2000): 53C27, 58J50, 58J60.     

Totally real embeddings of the torus intoC 2

In this short note we combine a construction of Viro and a result of Eliashberg and Harlamov to prove that there exist smooth totally real embeddings of the torus intoC ² which are isotopic but not so within the class of totally real surfaces. We also show how Viro's construction can be used to define an isotopy invariant for a certain class of complex curves inC P ².     

Martingale Representation of Functionals of Lévy Processes

Abstract

The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L ²(P), then from this definition a Clark–Ocone–Haussman type formula is derived. We also derive some explicit chaos expansions for some common functionals. Later we prove that the difference operator defined via the Fock space representation and the difference operator defined by Picard [Picard, J. Formules de dualitésur l'espace de Poisson. Ann. Inst. Henri Poincaré 1996, 32 (4), 509–548] are equal. Finally, we give an example of how the Clark–Ocone–Haussman formula can be used to solve a hedging problem in a financial market modelled by a Lévy process.     

The Dyadic Cesàro Operator on R+

The dyadic Cesàro operator C is introduced for functions in the space L ¹ := L ¹(R ₊) by means of the Walsh-Fourier transform defined by

Integrability of Analytic Almost Complex Structures on Banach Manifolds

We prove that the classical integrability condition for almost complex structures on finite-dimensional smooth manifolds also works in infinite dimensions in the case of almost complex structures that are real analytic on real analytic Banach manifolds. With this result at hand, we extend some known results concerning existence of invariant complex structures on homogeneous spaces of Banach–Lie groups. By way of illustration, we construct the complex flag manifolds associated with unital C^*-algebras.Mathematics Subject Classifications (2000): primary 32Q60; secondary 53C15, 58B12.     

Wentzell-Robin Boundary Conditions on C [0,1]

   Abstract. Given a∈ C ¹ [0,1], a(x)≥ α >0 , we prove that the second order differential operator on C[0,1] defined by A _W u:=(au')' with Wentzell-Robin boundary conditions

On a class of strongly hyperbolic systems

In this paper we prove a well-posedness result for the Cauchy problem. We study a class of first order hyperbolic differential [2] operators of rank zero on an involutive submanifold ofT ^* R ⁿ⁺¹-{0} and prove that under suitable assumptions on the symmetrizability of the lifting of the principal symbol to a natural blow up of the “singular part” of the characteristic set, the operator is strongly hyperbolic.     

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.     

On the injectivity of a class of Wiener-Hopf integral operators in a Hilbert space

A class of Wiener-Hopf integral operators, with kernels vanishing along the positive real axis, is obtained from considering weighted transaxial line-integrals of rotationally symmetric functions defined on ². An analysis of these operators is given when acting in, the Hilbert space L²(⁺). A necessary and sufficient condition for injectivity is established and inversion formulas are provided in some cases. A specific operator falling into this class, the so-called incomplete Abel transform., is presented and an inversion formula is given. This inversion formula makes precise a formal result previously established in Dallaset al. [J. Opt. Soc. Am. A4, 2039 (1987)] and it is also shown to be consistent with an inversion formula derived by Hansen [J. Opt. Soc. Am. A9, 2126 (1992)].This research was supported by NIH/NCI Grant R01 CA49261