Invertible sequences of bounded linear operators

In this paper, we study the invertibility of sequences consisting of finitely many bounded linear operators from a Hilbert space to others. We show that a sequence of operators is left invertible if and only if it is a g-frame. Therefore, our result connects the invertibility of operator sequences with frame theory.     

Spectral analysis of singular ordinary differential operators with indefinite weights

In this paper we develop a perturbation approach to investigate spectral problems for singular ordinary differential operators with indefinite weight functions. We prove a general perturbation result on the local spectral properties of selfadjoint operators in Krein spaces which differ only by finitely many dimensions from the orthogonal sum of a fundamentally reducible operator and an operator with finitely many negative squares. This result is applied to singular indefinite Sturm-Liouville operators and higher order singular ordinary differential operators with indefinite weight functions.     

Well-Boundedness of Sums and Products of Operators

A sufficient condition is given under which the sum, productand indeed any polynomial combination of a well-bounded operatorand a commuting real scalar-type spectral operator is well-bounded.This generalizes a result of Gillespie for Hilbert space operators.It is shown in particular that if X is a UMD space, then thesum of finitely many commuting real scalar-type spectral operatorsacting on X is a well-bounded operator (a result which failson general reflexive Banach spaces).     

Spectral Doubling of Normal Operators and Connections with Antiunitary Operators

Equations such as AB = B ^T A have been studied in the finite dimensional setting in (Linear Algebra Appl 369:279–294, 2003). These equations have implications for the spectrum of B, when A is normal. Our aim is to generalize these results to an infinite dimensional setting. In this case it is natural to use JB*J for some conjugation operator J in place of B ^T . Our main result is a spectral pairing theorem for a bounded normal operator B which is applied to the study of the equation KB = B*K for K an antiunitary operator. In particular, using conjugation operators, we generalize the notion of Hamiltonian operator and skew-Hamiltonian operator in a natural way, derive some of their properties, and give a characterization of certain operators B for which AB = (JB*J)A and BA = A(JB*J) and also those B with KB = B*K for certain antiunitary operators K.     

M-Canonical ideals in integral domains

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R     

Around operators not increasing the degree of polynomials

We present a generic operator J defined on the vectorial space of polynomial functions and we address the problem of finding the polynomial sequences that coincide with the (normalized) J-image of themselves. The technique developed assembles different types of operators and initiates with a transposition of the problem to the dual space. We provide examples for a J limited to three terms.     

Weyl's theorem and quasi-similarity

In this paper we give necessary and sufficient conditions of a finitely ascensive operator to obey Weyl's theorem and study Weyl's theorem for quasi-affine transforms. In particular, we give an improvement of the earlier result of Duggal forp-hyponormal operators.     

A remark on complete positivity of elementary operators

We show that many of the features of the theory of hypercyclic and supercyclic operators extend to that of finitely hypercyclic/supercyclic operators. In particular, subnormal operators, Banach space isometries, and thereforeC ₁ contractions are not finitely supercyclic.     

Relative frames on J-holomorphic curves

The aim of this work is to prove that any non-constant J-holomorphic disc with its boundary in a given Lagrangian submanifold can be decomposed in homology into a sum of finitely many J-holomorphic simple discs with the same Lagrangian boundary condition. As a consequence, in dimension higher than 6, any generic J-holomorphic disc is multicovered.     

Spectral analysis of doublyJ-nonexpanding operators

For a contracting operator in a space with an indefinite metric (i.e., for a doublyJ-nonexpanding operator) a characteristic operator-function is defined. On the basis of a detailed investigation of the properties of regular dilatations and characteristic functions of doublyJ-nonexpanding operators, a spectral analysis of these operators is carried out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 3, pp. 384–388, March, 1993.     

Monodromy-free Schrödinger operators with quadratically increasing potentials

We consider one-dimensional monodromy-free Schrödinger operators with quadratically increasing rational potentials. It is shown that all these operators can be obtained from the operator -?² + x² by finitely many rational Darboux transformations. An explicit expression is found for the corresponding potentials in terms of Hermite polynomials.     

Perturbations ofH-selfadjoint matrices,with applications to differential equations

In this paper we study perturbations of operators which are either selfadjoint or unitary with respect to an indefinite scalar product on a finite dimensional space (i.e.H-selfadjoint orH-unitary operators). The results allow us to describe systems of differential equations of higher order with selfadjoint coefficients which, together with all neighboring systems of the same kind, have only bounded solutions. An open problem concerning the structure of the connected components of such systems is posed.     

Directional maximal operators with smooth densities

We study directional maximal operators on ?ⁿ with smooth densities. We prove that if the classical directional maximal operator in a given set of directions is weak type (1, 1), then the corresponding smooth‐density maximal operator in that set of directions will be bounded on L^q for q suitably large, depending on the order of the stationary points of the density function. In contrast to the classical case, if q is too small, the smooth density operator need not be bounded on L^q. Improving upon previously known results, we also establish that if the density function has only finitely many extreme points, each of finite order, then any maximal operator in a finite sum of diadic directions is bounded on all L^q for q > 1 (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability of the Blok Theorem

W. Blok proved that varieties of boolean algebras with a single unary operator uniquely determined by their class of perfect algebras (i.e., duals of Kripke frames) are exactly those which are intersections of conjugate varieties of splitting algebras. The remaining ones share their class of perfect algebras with uncountably many other varieties. This theorem is known as the Blok dichotomy or the Blok alternative. We show that the Blok dichotomy holds when perfect algebras in the formulation are replaced by ω-complete algebras, atomic algebras with completely additive operators or algebras admitting residuals. We also generalize the Blok dichotomy for lattices of varieties of boolean algebras with finitely many unary operators. In addition, we answer a question posed by W. Dziobiak and show that classes of lattice-complete algebras or duals of Scott-Montague frames in a given variety are not determined by their subdirectly irreducible members. Received February 14, 2006; accepted in final form March 26, 2007.     

Spectral Components of Operators with Spectrum on a Curve

Trace class perturbations of normal operators with spectrum on a curve and spectral components of such operators are studied. We establish duality relations for the spectral components of an operator and its adjoint. The generalized Sz.-Nagy–Foia–Naboko functional model introduced in the paper is a basic tool for this theorem. The results have applications in nonself-adjoint scattering theory and to extreme factorizations of J-contraction-valued functions (J-inner-outer and A-regular-singular factorizations).     

On processes which cannot be distinguished by finite observation

A function J defined on a family C of stationary processes is finitely observable if there is a sequence of functions s _n such that s _n (x ₁,…, x _n ) → J(X) in probability for every process X=(x _n ) ∈ C. Recently, Ornstein and Weiss proved the striking result that if C is the class of aperiodic ergodic finite valued processes, then the only finitely observable isomorphism invariant defined on C is entropy [8]. We sharpen this in several ways. Our main result is that if X → Y is a zero-entropy extension of finite entropy ergodic systems and C is the family of processes arising from generating partitions of X and Y, then every finitely observable function on C is constant. This implies Ornstein and Weiss’ result, and extends it to many other families of processes, e.g., it follows that there are no nontrivial finitely observable isomorphism invariants for processes arising from the class of Kronecker systems, the class of mild mixing zero entropy systems, or the class of strong mixing zero entropy systems. It also follows that for the class of processes arising from irrational rotations, every finitely observable isomorphism invariant must be constant for rotations belonging to a set of full Lebesgue measure. This research was supported by the Israel Science Foundation (grant No. 1333/04)     

Determinacy of Bounded Complex Perturbations of Jacobi Matrices

We consider complex Jacobi matrices G which can be decomposed in the form G=J+C, where J is a real Jacobi matrix and C is a complex Jacobi matrix whose entries are uniformly bounded. We prove that the determinacy of the operator defined by G is equivalent to that of J. From this we deduce that the determinacy of G is equivalent to the coincidence between the domains of definition of the operators G and its adjoint G*.     

Reproducing kernels of generalized Sobolev spaces via a Green function approach with distributional operators

In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional operator P consisting of finitely or countably many distributional operators P _n , which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain appropriate full-space Green function G with respect to L := P ^*T P now becomes a conditionally positive function. In order to support this claim we ensure that the distributional adjoint operator P ^* of P is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space) associated with the Green function G can be embedded into or even be equivalent to a generalized Sobolev space. As an application, we take linear combinations of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant s _f,X to data values sampled from an unknown generalized Sobolev function f at data sites located in some set X ì \mathbbR^d{X \subset \mathbb{R}^d}. We provide several examples, such as Matérn kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert spaces of well-known reproducing kernels are equivalent to a generalized Sobolev space. These examples further illustrate how we can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the “best” kernel function for kernel-based approximation methods.     

Local rigidity and nonrigidity of symplectic pairs

It is shown that indefinite strictly almost K?hler and opposite K?hler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Drăghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraK?hler and opposite paraK?hler structures (\mathfrakJ,\mathfrakJ^￠){(\mathfrak{J},\mathfrak{J}^{\prime})} with \mathfrakJ{\mathfrak{J}} -invariant Ricci operator are shown.