Spectral Properties of Some Linear Matrix Differential Operators in Lp-spaces on $${\mathbb{R}}$$

A detailed study is made of matrix-valued, ordinary linear differential operators T in for 1 < p < ∞, which arise as the perturbation of a constant coefficient differential operator of order n ≥ 1 by a lower order differential operator which has a factorisation S = AB for suitable operators A and B. Via techniques from L ^p -harmonic analysis, perturbation theory and local spectral theory, it is shown that T satisfies certain local resolvent estimates, which imply the existence of local functional calculi and decomposability properties of T.      

Norm Attaining Operators and Pseudospectrum

It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓ_p-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥  > 1 and I + T does not attain its norm. The author would like to thank E. Shargorodsky for his interest and comments.     

Generalized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications

We study generalized polar decompositions of densely defined closed linear operators in Hilbert spaces and provide some applications to relatively (form) bounded and relatively (form) compact perturbations of self-adjoint, normal, and m-sectorial operators. Based upon work partially supported by the US National Science Foundation under Grant Nos. DMS-0400639 and FRG-0456306, and the Austrian Science Fund (FWF) under Grant No. Y330.     

Hankel and Toeplitz Transforms on H 1: Continuity, Compactness and Fredholm Properties

We revisit the boundedness of Hankel and Toeplitz operators acting on the Hardy space H ¹ and give a new proof of the old result stating that the Hankel operator H _a is bounded if and only if a has bounded logarithmic mean oscillation. We also establish a sufficient and necessary condition for H _a to be compact on H ¹. The Fredholm properties of Toeplitz operators on H ¹ are studied for symbols in a Banach algebra similar to C + H ^∞ under mild additional conditions caused by the differences in the boundedness of Toeplitz operators acting on H ¹ and H ². The first author was partially supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) and by the Greek Research Program “Pythagoras 2” (75% European funds and 25 National funds). The second author was fully supported by the European Commission IHP Network “Harmonic Analysis and Related Problems” (Contract Number: HPRN-CT-2001-00273-HARP) while he visited the first author at the University of Crete and later by the Academy of Finland Project 207048.     

On Some Product of Two Unbounded Self-Adjoint Operators

We give a spectral analysis of some unbounded normal product HK of two self-adjoint operators H and K (which appeared in [7]) and we say why it is not self-adjoint even if the spectrum of one of the operators is sufficiently “asymmetric”. Then, we investigate the self-adjointness of KH (given it is normal) for arbitrary self-adjoint H and K by giving a counterexample and some positive results and hence finishing off with the whole question of normal products of self-adjoint operators (appearing in [1, 7, 12]). The author was supported in part by CNEPRU: B01820070020 (Ministry of Higher Education, Algeria).     

Spectral Radius of Refinement and Subdivision Operators with Power Diagonal Dilations

Two types of estimate for the spectral radius of the multivariate refinement operator with power diagonal dilations are presented. One type contains multiplicator norm of number matrices generated by the symbol of the corresponding operator and by specific subsets of repeating fractions. These subsets are used together with the little Fermat theorem to establish estimates that comprise integrals over tori of various dimensions. Moreover, we note certain classes of symbols when the exact value of the spectral radius of refinement operator can be found. For the spectral radius of subdivision operators point value estimates are established. Submitted: April 25, 2007. Accepted: November 5, 2007.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Elements of the Theory of Linear Volterra Operators in Banach Spaces

The new definition of Volterra operator introduced in [5] allows specification of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations.     

Compact and Finite Rank Perturbations of Closed Linear Operators and Relations in Hilbert Spaces

For closed linear operators or relations A and B acting between Hilbert spaces and the concepts of compact and finite rank perturbations are defined with the help of the orthogonal projections P _A and P _B in onto the graphs of A and B. Various equivalent characterizations for such perturbations are proved and it is shown that these notions are a natural generalization of the usual concepts of compact and finite rank perturbations. Sadly, our colleague and friend Peter Jonas passed away on July, 18th 2007.     

Quadratic Inequalities for Hilbert Space Operators

Properties of sets of solutions to inequalities of the form

Xia Spectrum for Some Class of Operators

In this paper, we introduce Xia spectra of n-tuples of operators satisfying |T ²| ≥ U|T ²|U* for the polar decomposition of T = U|T| and we extend Putnam’s inequality to these tuples [7]. This research is partially supported by Grant-in-Aid Research No.17540176.     

Seminorm Related to Banach-Saks Property and Real Interpolation of Operators

We introduce the arithmetic separation of a sequence—a geometric characteristic for bounded sequences in a Banach space which describes the Banach-Saks property. We define an operator seminorm vanishing for operators with the Banach-Saks property. We prove quantitative stability of the seminorm for a class of operators acting between l _p -sums of Banach spaces. We show logarithmically convex-type estimates of the seminorm for operators interpolated by the real method of Lions and Peetre.      

Spectralizable Operators

We introduce the notion of spectralizable operators. A closed operator A in a Hilbert space is called spectralizable if there exists a non-constant polynomial p such that the operator p(A) is a scalar spectral operator in the sense of Dunford. We show that such operators belongs to the class of generalized spectral operators and give some examples where spectralizable operators occur naturally. Vladimir Strauss gratefully acknowledges support by DFG, Grant No. TR 903/3-1.     

Resolvent Estimates for Operators Belonging to Exponential Classes

For a, α > 0 let E(a, α) be the set of all compact operators A on a separable Hilbert space such that s _n (A) = O(exp(-an^α)), where s _n (A) denotes the n-th singular number of A. We provide upper bounds for the norm of the resolvent (zI − A)⁻¹ of A in terms of a quantity describing the departure from normality of A and the distance of z to the spectrum of A. As a consequence we obtain upper bounds for the Hausdorff distance of the spectra of two operators in E(a, α).      

Representation of Contractive Solutions of a Class of Algebraic Riccati Equations as Characteristic Functions of Maximal Dissipative Operators

Let

Ideal-Triangularizability of Semigroups of Positive Operators

Let be a multiplicative semigroup of positive operators on a Banach lattice E such that every is ideal-triangularizable, i.e., there is a maximal chain of closed subspaces of E that consists of closed ideals invariant under S. We consider the question under which conditions the whole semigroup is simultaneously ideal-triangularizable. In particular, we extend a recent result of G. MacDonald and H. Radjavi. We also introduce a class of positive operators that contains all positive abstract integral operators when E is Dedekind complete.      

A Remark on Support of the Principal Function for Class A Operators

Let be an invertible class A operator such that . Then we show that , where gT is the principal function of T. Moreover, we show that if T is pure, then .     

Test Function Criteria for Hankel Operators

It is well known that there are classes of test functions such that a Hankel operator is bounded if and only if it is bounded on those functions. Criteria are derived which determine whether a Hankel operator is compact or belongs to a particular Schatten class, in terms of its action on those test functions.     

Unbounded Toeplitz Operators

This partly expository article develops the basic theory of unbounded Toeplitz operators on the Hardy space H ², with emphasis on operators whose symbols are not square integrable. Unbounded truncated Toeplitz operators on coinvariant subspaces of H ² are also studied. In memory of Paul R. Halmos