Multiplication operators and traces of commutators

Some new sufficient conditions are found on operators A, B in a Hilbert space, under which trace (AB–BA)=0.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskgo Instituta im. V. A. Steklova AN SSSR, Vol. 135, pp. 182–194, 1984.The author is profoundly grateful to V. I. Lomonosov, V. V. Peller, M. Z. Solomyak, and Yu. T. Turovskii for helpful discussions of the questions considered in the paper and for information on the results of [1, 8, 11, 14].     

Multilinear dyadic operators and their commutators

We obtain a generalized paraproduct decomposition of the pointwise product of two or more functions that naturally gives rise to multilinear dyadic paraproducts and Haar multipliers. We then study the boundedness properties of these multilinear operators and their commutators with dyadic BMO functions. We also characterize the dyadic BMO functions via the boundedness of (a) certain paraproducts, and (b) the commutators of multilinear Haar multipliers and paraproduct operators.     

Norms of commutators of self-adjoint operators

In this note, the estimate of norms of commutators of self-adjoint operators is established.     

Traces on pseudodifferential operators and sums of commutators

The aim of this paper is to show that various known characterizations of traces on classical pseudodifferential operators can actually be obtained by very elementary considerations on pseudodifferential operators, using only basic properties of these operators. Thereby, we give a unified treatment of the determinations of the space of traces (i) on ΨDOs of non-integer order or of regular parity-class, (ii) on integer order ΨDOs, (iii) on ΨDOs of non-positive orders in dimension ≥ 2, and (iv) on ΨDOs of non-positive orders in dimension 1.     

Representations of the additive group of a nuclear space in terms of self-adjoint operators

We prove theorems on integral representations of the additive group of a real nuclear space in terms of self-adjoint operators. We assume that algebraic relations are realized in a dense invariant set of integral vectors. Translated from Ukrainskii Matematicheskii Zhurnal, Vol.51, No. 2, pp. 271–274, February, 1999.     

Vectorial Hankel operators,commutators and related operators of the Schatten-von Neumann class.

The Hankel operators of the class on the spaces of vector-valued functions are characterized in terms of their symbols. Different applications are considered. It is proved that the averaging projection onto the block-Hankel matrices is bounded on for 1 < p < + . A criterion for an operator commuting with a C_oo contraction to be in is obtained in terms of the Sz.-Nagy-Foia lifting theorem. Finally the obtained results are applied to describe various commutators of the class.     

Compactness of the commutators of homogeneous singular integral operators

Let TΩ be the singular integral operator with kernel Ω(x)/|x|~n,where Ω is homogeneous of degree zero,integrable and has mean value zero on the unit sphere S~(n-1).In this paper,by Fourier transform estimates,Littlewood-Paley theory and approximation,the authors prove that if Ω∈L(lnL)~2(S~(n-1)),then the commutator generated by T_Ω and CMO(R~n) function,and the corresponding discrete maximal operator,are compact on L~p(R~n,|x|~(γp)) for p∈(1,∞) and γ_p ∈(-1,p-1).     

Weighted estimates for the multilinear commutators of the Littlewood-Paley operators

In this paper, weighted Lp estimates and sharp weighted endpoint estimates for the mul-tilinear commutators of the Littlewood-Paley operators are established.     

On a certain type of commutators of operators

LetH be a separable infinite-dimensional Hilbert space and letC be a normal operator andG a compact operator onH. It is proved that the following four conditions are equivalent.

1. C +G is a commutatorAB-BA with self-adjointA.
2. There exists an infinite orthonormal sequencee _j inH such that |Σ _j ⁿ =1 (Ce _j, e_j)| is bounded.
3. C is not of the formC ₁ ⊕C ₂ whereC ₁ has finite dimensional domain andC ₂ satisfies inf {|(C ₂ x, x)|: ‖x‖=1}>0.
4. 0 is in the convex hull of the set of limit points of spC.

     

Norm inequalities for commutators of positive operators and applications

Let X, Y, and Z be operators on a Hilbert space such that X and Z are positive. It is shown that

Characterization for commutators of n-dimensional fractional Hardy operators

In this paper, it was proved that the commutator Hβ,b generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from Lp1(Rn) to Lp2 (Rn) if and only if b is a C(M)O(Rn) function, where 1/p1 - 1/p2 = β/n, 1 ＜ p1 ＜∞, 0 ≤β＜ n. Furthemore,the characterization of Hβ,b on the homogenous Herz space (K)qα,p(Rn) was obtained.     

Rigidity of commutators and elementary operators on Calkin algebras

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ ^p ) ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ ^p ), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ ^p )≡L(ℓ ^p )/K(ℓ ^p ); , where quotient elements are denoted bys=S+K(ℓ ^p ) forSεL(ℓ ^p ). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ ^p ) whenever ε _a,b fails to be one-one, while the quotient is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces. Supported by the Academy of Finland, Project 32837.     

New formula for ln(eAeB) in terms of commutators of A and B

We establish the formula $$\ln (e^B e^A ) = \smallint _0^t \psi (e^{ - \tau ad_A } e^{ - \tau ad_B } ) e^{ - \tau ad_A } d\tau (A + B),$$ where Ψ(x)=(In x)/(x ? 1); here A and B are elements of a. finite-dimensional Lie algebra which satisfy certain conditions. This formula enables us, in particular, to give a simple proof of the Campbell-Hausdorff theorem. We also give a generalization of the formula to the case of an arbitrary number of factors.