On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Harris operators

A method is constructed which leads to a proof for both the “zero-two” law, and the Ornstein-Métivier-Brunel Theorem for Harris operators. For the proof it is not necessary to assume that the measure space is measurable and the operator need not be given by a transition probability. We strove to make these notes self-contained.     

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.     

Elementary operators on self-adjoint operators

Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that and are surjective maps such that M(AM^∗(B)A)=M(A)BM(A) and M^∗(BM(A)B)=M^∗(B)AM^∗(B) for every pair A∈As, B∈Bs. Then there exist an invertible bounded linear or conjugate-linear operator and a constant c∈{−1,1} such that M(A)=cTAT^∗, A∈As, and M^∗(B)=cT^∗BT, B∈Bs.     

Singular integral operators on complicated contours and pseudodifferential operators

We apply the pseudodifferential operator technique to the study of algebras of singular operators on complicated contours. This technique is used to construct a symbolic calculus for theC ^*-algebra generated by singular integral operators whose coefficients may have singularities of the second kind on complicated contours; the curves forming a node are not required to have a tangent at the node. Translated fromMatematicheskie Zametki, Vol. 58, No. 1, pp. 67–85, July, 1995. This work was partially supported by the Russian Foundation for Basic Research under grant No. 93-04-691.     

Hankel operators in the set of essential Toeplitz operators

In this paper, we study the Hankel operators belonging toE which denotes all of the essential Toeplitz operators, and show that the class of symbols of these Hankel operators is a Douglas algebra. Meanwhile, using the concept of essential commutant and some knowledge of the theory of function algebras, we solve a problem raised by Josè Barria and P. R. Halmos in [1].     

Universal non-completely-continuous operators

A bounded linear operator between Banach spaces is calledcompletely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators fromL ₁ into an arbitrary Banach space, namely, the operator fromL ₁ into ⊆_∞ defined byT ₀(f) = (∫r _n f d μ) _n>-0, wherer _n is thenth Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach spaces. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space. Supported in part by NSF grant DMS-9306460. Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University (supported in part by NSF grant DMS-9311902). Supported in part by NSF grant DMS-9003550.     

On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H² of the unit disk. We show that if is of finite rank for every inner function θ, then X=T_?+F for some Toeplitz operator T_? and some finite rank operator F on H². This solves a variant of an open question where the compactness replaces the finite rank conditions.     

Pseudo-differential operators

Summary We present here a number of results on some aspects of Kohn-Nirenberg's theory of pseudo-differential operators. We hope that some parts of Kohn-Nirenberg's paper[1] are presented here in a more detailed and explicit form; this could help a larger audience to understand their ideas and methods. Entrata in Redazione il 3 giugno 1971. This research is supported by a grant of the N.R.C. of Canada.     

Central Adams operators

This is a supplement to our analysis [2] of Adams, or V₁, periodicity operators on mod p^a-Moore spaces using the J-map defined by a chosen solution of the Adams conjecture. Motivated by recent work of E. Devinatz, M. Hopkins and J. Smith, we use the same method to construct natural periodicity operators for arbitrary torsion spaces.     

Testing linear operators

The goal of testing is to determine whether an implementation linear operatorA conforms to a specification linear operatorS within a given error bound for all elements from an input setF. Suppose that an upper boundK on the norm of the difference ofS andA is given a priori. Then it is shown that in general any finite number of tests is inconclusive both in the worst case and on the average. However, the testing problem is still decidable in the limit for an arbitraryK; there is an algorithm of an infinite sequence of test-and-guess such that all but finitely many guesses are correct. On the other hand, if the error bound is relaxed for weak conformance then finite tests suffice even in the worst case and tight lower and upper bounds on the number of tests are derived. The test set is universal; it only depends on the set of valid inputsF. Furthermore, the test elements are on the boundary ofF. Two examples are used to illustrate the approaches and the paper is concluded with comments on two related problems: computation and verification.This work was done while consulting at AT&T Bell Laboratories, and is partially supported by the National Science Foundation grant IRI-92-12597 and the Air Force Office of Scientific Research 91-0347.     

Perturbing Fredholm operators to obtain invertible operators

A condition is given on a set $\text{Ol}$ of operators on Hilbert space that guarantees it has the following property: For any Fredholm operator T of index zero there exists anA?$\text{A}$ such that T + ?A is invertible for all sufficiently small nonzero ?. As a corollary one obtains in a quite general setting the density of the invertible Toeplitz operators in the set of Fredholm Toeplitz operators of index zero.     

Testing nonlinear operators

We study testing of nonlinear operators; we want to test whether an implementation operator conforms to a specification operator. The problem is difficult, since there can be infinitely many possible inputs but we can only test finitely many of them. An implementation operator may perform well on the tested inputs but may be faulty on the untested inputs. In general, finite testing is inherently inconclusive. Consequently, we modify the problem in three different directions and obtain positive results: (1) We consider an infinite sequence of tests and prove that testing is decidable in the limit; (2) We relax the error criterion and show that finite testing is conclusive, however, the cost could be formidable; and (3) We tolerate faults on a negligible subset of inputs and develop a probabilistic testing algorithm with a significantly reduced cost. Our results indicate that test sets are universal; they only depend on the structure of the input set. In fact, they are provided by an net of the input set.This work was done while consulting at AT&T Bell Laboratories, and is partially supported by the National Science Foundation grant IRI-89-07215 and the Air Force Office of Scientific Research 91-0347.     

Supersymmetric Dirichlet operators

A canonical supersymmetric extension of classical Dirichlet forms and operators given by measures on linear spaces is considered. In the case of log-concave measures, the supersymmetric Dirichlet operator generates a hypercontractive semigroup. This setting is appropriate for infinite-dimensional geometry and quantum field theory.Dedicated to Yu. M. Berezanskii on the occasion of his seventieth birthdayPublished in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 583–592, May, 1995.The present work was supported by the DFG.     

Regular closure operators

In an E,M-categoryX for sinks, we identify necessary conditions for Galois connections from the power collection of the class of (composable pairs) of morphisms inM to factor through the lattice of all closure operators onM, and to factor through certain sublattices. This leads to the notion ofregular closure operator. As one byproduct of these results we not only arrive (in a novel way) at the Pumplün-Röhrl polarity between collections of morphisms and collections of objects in such a category, but obtain many factorizations of that polarity as well. (One of these factorizations constituted the main result of an earlier paper by the same authors). Another byproduct is the clarification of the Salbany construction (by means of relative dominions) of the largest idempotent closure operator that has a specified class ofX-objects as separated objects. The same relation that is used in Salbany's relative dominion construction induces classical regular closure operators as described above. Many other types of closure operators can be obtained by this technique; particular instances of this are the idempotent and modal closure operators that in a Grothendieck topos correspond to the Grothendieck topologies.Dedicated to Professor Dieter Pumplün, on his 60th birthdayResearch partially supported by the Faculty of Arts and Sciences, University of Puerto Rico, Mayagüez Campus during a sabbatical visit at Kansas State University.     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Equivalent composition operators

If is an analytic function taking the unit disk into itself then the composition operatorC can be defined on the Hardy spaceH ^p(D) byC (f)=fo. In this work, the question of when two of these operators are equivalent in some sense is studied. In some cases, it turns out that the only equivalences are those that are induced by an invertible composition operator. However, other cases are exhibited in which there are equivalences that are not induced by an invertible composition operator.Much of this work comes from the author's thesis, written under the direction of Carl C. Cowen at Purdue University. Final preparations were accomplished with the help of a faculty development grant from the University of Tampa.     

Faber-hypercyclic operators

Let Ω be a bounded domain of the complex plane whose boundary is a closed Jordan curve and (F _n ) _n≥0 the sequence of Faber polynomials of Ω. We say that a bounded linear operator T on a separable Banach space X is Ω-hypercyclic if there exists a vector x of X such that {F _n (T)x: n ≥ 0} is dense in X. We show that many of the results in the spectral theory of hypercyclic operators involving the unit disk or its boundary have Ω-hypercyclic counterparts which involve the domain Ω or its boundary. The influence of the geometry of Ω or the smoothness of its boundary on Faber-hypercyclicity is also discussed.