Generalized Fredholm property and a priori estimates for linear operators on tensor products of Hilbert spaces

For a linear operator acting in a Hilbert space, the generalized Fredholm property (invertibility modulo a certain ideal) is proved to be equivalent to certaina priori estimates. This result is applied to establish a connection between properties of linear operators on tensor products of Hilbert spaces, such asn- andd-normality, the (generalized and ordinary) Fredholm property, and appropriatea priori estimates.Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 902–912, December, 1998.The author is grateful to V. M. Deundyak for useful discussion of this work.This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01195.     

Singular Integral Operators with Coefficients of a Special Structure Related to Operator Equalities

In our previous works we have constructed operator equalities which transform scalar singular integral operators with shift to matrix characteristic singular integral operators without shift and found some of their applications to problems with shift. In this article the operator equalities are used for the study of matrix characteristic singular integral operators. Conditions for the invertibility of the singular integral operators with orientation preserving shift and coefficients with a special structure generated by piecewise constant functions, t, t ⁻¹, were found. Conditions for the invertibility of the matrix characteristic singular integral operators with four-valued piecewise constant coefficients of a special structure were likewise obtained. Submitted: June 15, 2007. Revised: October 25, 2007. Accepted: November 5, 2007.     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

The invertibility of convolution type operators on a union of intervals and the corona theorem

The invertibility of convolution type operators on unions of intervals is studied. Sufficient conditions of invertibility for some classes of these operators are established. Solvability results forn-term corona problems are obtained using two different approaches: one involving reduction ton–1 Riemann-Hilbert problems in two variables and another involving reduction to two-term corona problems. The invertibility of the convolution operators on a union of intervals is also related to the invertibility of associated convolution operators on single intervals. Formulas for the inverse operators are given.Work partially supported by F. C. T. (Portugal) Project Praxis XXI/2/2.1/MAT/441/94 and by CONACYT (México) grant, Cátedra Patrimonial, No. 990017-EX.     

Lp-Approximable Sequences of Vectors and Limit Distribution of Quadratic Forms of Random Variables

The properties of L₂-approximable sequences established here form a complete toolkit for statistical results concerning weighted sums of random variables, where the weights are nonstochastic sequences approximated in some sense by square-integrable functions and the random variables are “two-wing” averages of martingale differences. The results constitute the first significant advancement in the theory of L₂-approximable sequences since 1976 when Moussatat introduced a narrower notion of L₂-generated sequences. The method relies on a study of certain linear operators in the spaces L_p and l_p. A criterion of L_p-approximability is given. The results are new even when the weight generating function is identically 1. A central limit theorem for quadratic forms of random variables illustrates the method.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.     

Mapping Properties of Some Singular Operators in Besov Type Subspaces of C( − 1, 1)

We study mapping properties, boundedness and invertibility of some Cauchy singular integral operators in a scale of pairs of Besov type subspaces of C( − 1, 1). Our results include all those already available in the literature.     

Invertible Weak Entwining Structures and Weak C-cleft Extensions

In this note we improve the definition of invertible weak entwining structure. Also we clarify what invertibility means for weak cleft extensions and weak C-Galois extensions.      

Some connections between an operator and its Helton class

In this paper we show that the nilpotent perturbation of operators in the Helton class of p-hyponormal operators has scalar extensions. As a corollary we get that the nilpotent perturbation of each operator in the Helton class of p-hyponormal operators has a nontrivial invariant subspace if its spectrum has nonempty interior in the plane.     

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.     

-Operator frames for a Banach space

In this paper, (p,Y)-Bessel operator sequences, operator frames and (p,Y)-Riesz bases for a Banach space X are introduced and discussed as generalizations of the usual concepts for a Hilbert space and of the g-frames. It is proved that the set of all (p,Y)-Bessel operator sequences for a Banach space X is a Banach space and isometrically isomorphic to the operator space B(X,ℓ^p(Y)). Some necessary and sufficient conditions for a sequence of operators to be a (p,Y)-Bessel operator sequence are given. Also, a characterization of an independent (p,Y)-operator frame for X is obtained. Lastly, it is shown that an independent (p,Y)-operator frame for X is just a (p,Y)-Riesz basis for X and has a unique dual (q,Y^*)-operator frame for X^*.     

Greedy Bases Are Best for m-Term Approximation

We study the approximation of a subset K in a Banach space X by choosing first basis B and then using n-term approximation. Into the competition for best bases we enter all unconditional bases for X. We show that if the subset K⊂X is well aligned with respect to a greedy basis B, then, in a certain sense, this basis is the best for this type of approximation. Our result improves the recent result of DeVore, Petrova and Temlyakov.      

Very smooth points of spaces of operators

In this paper we study very smooth points of Banach spaces with special emphasis on spaces of operators. We show that when the space of compact operators is anM-ideal in the space of bounded operators, a very smooth operatorT attains its norm at a unique vectorx (up to a constant multiple) andT(x) is a very smooth point of the range space. We show that if for every equivalent norm on a Banach space, the dual unit ball has a very smooth point then the space has the Radon-Nikodym property. We give an example of a smooth Banach space without any very smooth points.     

Invertibility Conditions for Second-Order Differential Operators in the Space of Continuous Bounded Functions

We prove the invertibility of second-order differential operators with constant operator coefficients acting on the Banach space of bounded continuous functions on the real line under the condition that they are uniformly injective (in particular, left invertible) or surjective (in particular, right invertible). We show that if these operators are considered on the space of periodic functions, then the unilateral invertibility does not imply the invertibility of such operators. We obtain criteria for the injectivity, surjectivity, and invertibility of differential operators on the space of periodic functions.     

Operators on Differential Form Spaces for Riemann Surfaces

In the present paper, a problem of Ioana Mihaila is negatively answered on the invertibility of composition operators on Riemann surfaces, and it is proved that the composition operator Cp is Predholm if and only if it is invertible if and only if p is invertible for some special cases. In addition, the Toeplitz operators on ∧1 2, a（M） for Riemann surface M are defined and some properties of these operators are discussed.     

Weak-open compact images of metric spaces

The main results of this paper are that (1) a space X is g-developable if and only if it is a weak-open π image of a metric space, one consequence of the result being the correction of an error in the paper of Z. Li and S. Lin (2) characterizations of weak-open compact images of metric spaces, which is another answer to a question in in the paper of Y. Ikeda, C. liu and Y. Tanaka.     

On subnormal operators

LetS be a pure subnormal operator such thatC*(S), theC*-algebra generated byS, is generated by a unilateral shiftU of multiplicity 1. We obtain conditions under which 5 is unitarily equivalent toα + βU, α andβ being scalars orS hasC*-spectral inclusion property. It is also proved that if in addition,S hasC*-spectral inclusion property, then so does its dualT andC*(T) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.     

Interior and closure operators on bounded residuated lattice ordered monoids

GMV-algebras endowed with additive closure operators or with its duals-multiplicative interior operators (closure or interior GMV-algebras) were introduced as a non-commutative generalization of topological Boolean algebras. In the paper, the multiplicative interior and additive closure operators on DRl-monoids are introduced as natural generalizations of the multiplicative interior and additive closure operators on GMV-algebras.