Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces,H ^s,p , which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders:S – 1/p . For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.This research was supported by Junta Nacional de Investigação Científica e Tecnológica (Portugal) and the Bundesminister für Forschung und Technologie (Germany) within the projectSingular Operators-new features and applications, and by a PRAXIS XXI project under the titleFactorization of Operators and Applications to Mathematical Physics.     

Transitive Spaces of Operators

We investigate algebraic and topological transitivity and, more generally, k-transitivity for linear spaces of operators. In finite dimensions, we determine minimal dimensions of k-transitive spaces for every k, and find relations between the degree of transitivity of a product or tensor product on the one hand and those of the factors on the other. We present counterexamples to some natural conjectures. Some infinite dimensional analogues are discussed. A simple proof is given of Arveson’s result on the weak-operator density of transitive spaces that are masa bimodules. Authors partially supported by NSERC grants.     

UBIQUITOUS PROPERTIES OF CERTAIN OPERATIONAL QUANTITIES OF LINEAR RELATIONS IN NORMED SPACES

Abstract

Let X and Y be normed spaces and T a linear relation in X x Y. Certain known operational quantities of linear operators are extended to linear relations. Properties of T such as precompactness, strict singularity, upper semi-Fredholmness and partial continuity are defined and characterised in terms of the given quantities, and certain uniform boundedness properties of these quantities derived.     

Some inequalities for the Euclidean operator radius of two operators in Hilbert spaces

Some sharp bounds for the Euclidean operator radius of two bounded linear operators in Hilbert spaces are given. Their connection with Kittaneh’s recent results which provide sharp upper and lower bounds for the numerical radius of linear operators are also established.     

Mixed Boundary Value Problems for the Helmholtz Equation in a Quadrant

The main objective is the study of a class of boundary value problems in weak formulation where two boundary conditions are given on the half-lines bordering the first quadrant that contain impedance data and oblique derivatives. The associated operators are reduced by matricial coupling relations to certain boundary pseudodifferential operators which can be analyzed in detail. Results are: Fredholm criteria, explicit construction of generalized inverses in Bessel potential spaces, eventually after normalization, and regularity results. Particular interest is devoted to the impedance problem and to the oblique derivative problem, as well.     

A Theorem of Beurling-sLax Type for Hilbert Spaces of Functions Analytic in the Unit Ball

Schur multipliers on the unit ball are operator-valued functions for which the N-variable Schwarz-Pick kernel is nonnegative. In this paper, the coefficient spaces are assumed to be Pontryagin spaces having the same negative index. The associated reproducing kernel Hilbert spaces are characterized in terms of generalized difference-quotient transformations. The connection between invariant subspaces and factorization is established.     

The core operator and congruent submodules

The Hardy spaces H²(D²) can be conveniently viewed as a module over the polynomial ring C[z₁,z₂]. Submodules of H²(D²) have connections with many areas of study in operator theory. A large amount of research has been carried out striving to understand the structure of submodules under certain equivalence relations. Unitary equivalence is a well-known equivalence relation in set of submodules. However, the rigidity phenomenon discovered in [Douglas et al., Algebraic reduction and rigidity for Hilbert modules, Amer. J. Math. 117 (1) (1995) 75-92] and some other related papers suggests that unitary equivalence, being extremely sensitive to perturbations of zero sets, lacks the flexibility one might need for a classification of submodules. In this paper, we suggest an alternative equivalence relation, namely congruence. The idea is motivated by a symmetry and stability property that the core operator possesses. The congruence relation effectively classifies the submodules with a finite rank core operator. Near the end of the paper, we point out an essential connection of the core operator with operator model theory.     

The Virozub–Matsaev Condition and Spectrum of Definite Type for Self-adjoint Operator Functions

We establish sufficient conditions for the so-called Virozub–Matsaev condition for twice continuously differentiable self-adjoint operator functions. This condition is closely related to the existence of a local spectral function and to the notion of positive type spectrum. Applications to self-adjoint operators in Krein spaces and to quadratic operator polynomials are given. Received: September 22, 2007. Accepted: September 29, 2007.     

A reproducing kernel thesis for operators on Bergman-type function spaces

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, the unit polydisc and more generally to weighted Fock spaces.     

On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.     

On the filling in holes problem for operator matrices

We consider upper-triangular 2-by-2 operator matrices and are interested in the set that has to be added to certain spectra of the matrix in order to get the union of the corresponding spectra of the two diagonal operators. We show that in the cases of the Browder essential approximate point spectrum, the upper semi-Fredholm spectrum, or the lower semi-Fredholm spectrum the set in question need not to be an open set but may be just a singleton. In addition, we modify and extend known results on Hilbert space operators to operators on Banach spaces.     

The isomorphism problem for some universal operator algebras

This paper addresses the isomorphism problem for the universal (non-self-adjoint) operator algebras generated by a row contraction subject to homogeneous polynomial relations. We find that two such algebras are isometrically isomorphic if and only if the defining polynomial relations are the same up to a unitary change of variables, and that this happens if and only if the associated subproduct systems are isomorphic. The proof makes use of the complex analytic structure of the character space, together with some recent results on subproduct systems. Restricting attention to commutative operator algebras defined by a radical ideal of relations yields strong resemblances with classical algebraic geometry. These commutative operator algebras turn out to be algebras of analytic functions on algebraic varieties. We prove a projective Nullstellensatz connecting closed ideals and their zero sets. Under some technical assumptions, we find that two such algebras are isomorphic as algebras if and only if they are similar, and we obtain a clear geometrical picture of when this happens. This result is obtained with tools from algebraic geometry, reproducing kernel Hilbert spaces, and some new complex-geometric rigidity results of independent interest. The C^?-envelopes of these algebras are also determined. The Banach-algebraic and the algebraic classification results are shown to hold for the wot-closures of these algebras as well.     

On simultaneous triangularization of collections of compact operators

In this paper we consider collections of compact (resp. C_p class) operators on arbitrary Banach (resp. Hilbert) spaces. For a subring R of reals, it is proved that an R-algebra of compact operators with spectra in R on an arbitrary Banach space is triangularizable if and only if every member of the algebra is triangularizable. It is proved that every triangularizability result on certain collections, e.g., semigroups, of compact operators on a complex Banach (resp. Hilbert) space gives rise to its counterpart on a real Banach (resp. Hilbert) space. We use our main results to present new proofs as well as extensions of certain classical theorems (e.g., those due to Kolchin, McCoy, and others) on arbitrary Banach (resp. Hilbert) spaces.     

Regular Functions of Operators on Tensor Products of Hilbert Spaces

A class of linear operators on tensor products of Hilbert spaces is considered. Estimates for the norm of operator-valued functions regular on the spectrum are derived. These results are new even in the finite-dimensional case. By virtue of the obtained estimates, we derive stability conditions for semilinear differential equations. Applications of the mentioned results to integro-differential equations are also discussed.     

Inertia theorems for operator pencils and applications

In this paper we obtain general infinite dimensional inertia theorems for linear pencils in Hilbert space which cover previously known results for the finite dimensional case and for block weighted shifts. Connections with definite subspaces for contractions in spaces with indefinite metric are discussed.     

Recurrence relations for a Newton-like method in Banach spaces

The convergence of iterative methods for solving nonlinear operator equations in Banach spaces is established from the convergence of majorizing sequences. An alternative approach is developed to establish this convergence by using recurrence relations. For example, the recurrence relations are used in establishing the convergence of Newton's method [L.B. Rall, Computational Solution of Nonlinear Operator Equations, Robert E. Krieger, New York, 1979] and the third order methods such as Halley's, Chebyshev's and super Halley's [V. Candela, A. Marquina, Recurrence relations for rational cubic methods I: the Halley method, Computing 44 (1990) 169–184; V. Candela, A. Marquina, Recurrence relations for rational cubic methods II: the Halley method, Computing 45 (1990) 355–367; J.A. Ezquerro, M.A. Hernández, Recurrence relations for Chebyshev-type methods, Appl. Math. Optim. 41 (2000) 227–236; J.M. Gutiérrez, M.A. Hernández, Third-order iterative methods for operators with bounded second derivative, J. Comput. Appl. Math. 82 (1997) 171–183; J.M. Gutiérrez, M.A. Hernández, Recurrence relations for the Super–Halley method, Comput. Math. Appl. 7(36) (1998) 1–8; M.A. Hernández, Chebyshev's approximation algorithms and applications, Comput. Math. Appl. 41 (2001) 433–445 [10]].     

Quasi-Free Resolutions Of Hilbert Modules

The notion of a quasi-free Hilbert module over a function algebra $\mathcal{A}$ consisting of holomorphic functions on a bounded domain $\Omega$ in complex m space is introduced. It is shown that quasi-free Hilbert modules correspond to the completion of the direct sum of a certain number of copies of the algebra $\mathcal{A}$. A Hilbert module is said to be weakly regular (respectively, regular) if there exists a module map from a quasi-free module with dense range (respectively, onto). A Hilbert module $\mathcal{M}$ is said to be compactly supported if there exists a constant $\beta$ satisfying $\|\varphi f\| \leq \beta \ |\varphi \| \textsl{X} \|f \|$ for some compact subset X of $\Omega$ and $\varphi$ in $\mathcal{A}$, f in $\mathcal{M}$. It is shown that if a Hilbert module is compactly supported then it is weakly regular. The paper identifies several other classes of Hilbert modules which are weakly regular. In addition, this result is extended to yield topologically exact resolutions of such modules by quasi-free ones.     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell     

A new approach to spectral approximation

A new technique for approximating eigenvalues and eigenvectors of a self-adjoint operator is presented. The method does not incur spectral pollution, uses trial spaces from the form domain, has a self-adjoint algorithm, and exhibits superconvergence.     

ON ORBITS OF POLYNOMIAL MAPS IN BANACH SPACES

Abstract

We obtain some results concerning the notions of hypercyclicity and supercyclicity for continuous homogeneous polynomial maps in Banach spaces. We also present a generalization of a theorem due to Beauzamy concerning the existence of orbits tending to infinity for continuous linear operators with spectral radius > 1.