Tent spaces associated with semigroups of operators

We study tent spaces on general measure spaces (Ω,μ). We assume that there exists a semigroup of positive operators on Lp(Ω,μ) satisfying a monotone property but do not assume any geometric/metric structure on Ω. The semigroup plays the same role as integrals on cones and cubes in Euclidean spaces. We then study BMO spaces on general measure spaces and get an analogue of Fefferman's H¹-BMO duality theory. We also get a H¹-BMO duality inequality without assuming the monotone property. All the results are proved in a more general setting, namely for noncommutative Lp spaces.     

Operators with connected spectrum + compact operators = strongly irreducible operators

Herrero’s conjecture that each operator with connected spectrum acting on complex, separable Hilbert space can be written as the sum of a strongly irreducible operator and a compact operator is proved. Jiang, C. L., Power, S., Wang, Z. Y., Biquasitriangular + small compact = strongly irreducible,J. London Math., to be published.     

Radical of weakly ordered semigroup algebras

We define the notion of weakly ordered semigroups. For this class of semigroups, we compute the radical of the semigroup algebras. This generalizes some results on left regular bands and on 0- Hecke algebras.     

Pseudo-differential operators and Markov semigroups on compact Lie groups

We extend the Ruzhansky-Turunen theory of pseudo-differential operators on compact Lie groups into a tool that can be used to investigate group-valued Markov processes in the spirit of the work in Euclidean spaces of N. Jacob and collaborators. Feller semigroups, their generators and resolvents are exhibited as pseudo-differential operators and the symbols of the operators forming the semigroup are expressed in terms of the Fourier transform of the transition kernel. The symbols are explicitly computed for some examples including the Feller processes associated to stochastic flows arising from solutions of stochastic differential equations on the group driven by Lévy processes. We study a family of Lévy-type linear operators on general Lie groups that are pseudo-differential operators when the group is compact and find conditions for them to give rise to symmetric Dirichlet forms.     

Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.     

Spectral representation of local symmetric semigroups of operators

A spectral integral representation is established for locally defined symmetric semigroups of operators, with indices which are not restricted to a neighborhood of zero. This extends the well-known results of Fröhlich (Adv. Appl. Math. 1:237–256, 1980) and Klein and Landau (J. Funct. Anal. 44:121–137, 1981).     

Spectral conditions on Lie and Jordan algebras of compact operators

We investigate the properties of bounded operators which satisfy a certain spectral additivity condition, and use our results to study Lie and Jordan algebras of compact operators. We prove that these algebras have nontrivial invariant subspaces when their elements have sublinear or submultiplicative spectrum, and when they satisfy simple trace conditions. In certain cases we show that these conditions imply that the algebra is (simultaneously) triangularizable.     

Hyper-Tauberian algebras and weak amenability of Figà-Talamanca-Herz algebras

We study certain commutative regular semisimple Banach algebras which we call hyper-Tauberian algebras. We first show that they form a subclass of weakly amenable Tauberian algebras. Then we investigate the basic and hereditary properties of them. Moreover, we show that if A is a hyper-Tauberian algebra, then the linear space of bounded derivations from A into any Banach A-bimodule is reflexive. We apply these results to the Figà-Talamanca-Herz algebra A_p(G) of a locally compact group G for p∈(1,∞). We show that A_p(G) is hyper-Tauberian if the principal component of G is abelian. Finally, by considering the quantization of these results, we show that for any locally compact group G, A_p(G), equipped with an appropriate operator space structure, is a quantized hyper-Tauberian algebra. This, in particular, implies that A_p(G) is operator weakly amenable.     

Flags in zero dimensional complete intersection algebras and indices of real vector fields

We introduce bilinear forms in a flag in a complete intersection local -algebra of dimension 0, related to the Eisenbud–Levine, Khimshiashvili bilinear form. We give a variational interpretation of these forms in terms of Jantzen’s filtration and bilinear forms. We use the signatures of these forms to compute in the real case the constant relating the GSV-index with the signature function of vector fields tangent to an even dimensional hypersurface singularity, one being topologically defined and the other computable with finite dimensional commutative algebra methods. Partially supported by Plan Nacional I+D grant no. MTM2004-07203-C02-02, Spain and CONACYT 40329, México.     

Gradient systems of closed operators

A classical result on the existence of global attractors for gradient systems is extended to the case of a semigroup S(t) lacking strong continuity, but satisfying the weaker property of being a closed map for every fixed t ≥ 0.      

An application of semigroups of locally Lipschitz operators to Carrier equations with acoustic boundary conditions

A generation theorem of semigroups of locally Lipschitz operators on a subset of a real Banach space is given and applied to the problem of the well-posedness of the Carrier equation utt−κ(‖u^‖2)Δu+γ|ut|^p−1ut=0 in Ω×(0,∞) with acoustic boundary condition, where p>2 and Ω is a bounded domain in an arbitrary dimensional space.     

On the sum of a narrow and a compact operators

Our main technical tool is a principally new property of compact narrow operators which works for a domain space without an absolutely continuous norm. It is proved that for every Köthe F-space X and for every locally convex F-space Y   the sum _T1+_T2$T_1+T_2$ of a narrow operator _T1:X→Y$T_1:X\to Y$ and a compact narrow operator _T2:X→Y$T_2:X\to Y$ is a narrow operator. This gives a positive answers to questions asked by M. Popov and B. Randrianantoanina [6, Problems 5.6 and 11.63].     

Diagonalization of compact operators on Hilbert modules overC *-algebras of real rank zero

The classical Hilbert-Schmidt theorem can be extended to compact operators on HilbertA-modules overW ^*-algebras of finite type; i.e., with minor restrictions, compact operators onH* _A can be diagonalized overA. We show that ifB is a weakly denseC ^*-subalgebra ofA with real rank zero and if some additional condition holds, then the natural extension fromH _B toH* _A ⊃H _B of a compact operator can be diagonalized so that the diagonal elements belong to the originalC ^*-algebraB. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 865–870, December, 1997. Translated by O. V. Sipacheva     

The preservation of Sahlqvist equations in completions of Boolean algebras with operators

Monk [1970] extended the notion of the completion of a Boolean algebra to Boolean algebras with operators. Under the assumption that the operators of such an algebra are completely additive, he showed that the completion of always exists and is unique up to isomorphisms over . Moreover, strictly positive equations are preserved under completions a strictly positive equation that holds in must hold in the completion of . In this paper we extend Monk’s preservation theorem by proving that certain kinds of Sahlqvist equations (as well as some other types of equations and implications) are preserved under completions. An example is given that shows that arbitrary Sahlqvist equations need not be preserved. Received May 3, 1998; accepted in final form October 7, 1998.     

Automorphisms of the algebra ofq-difference operators

In this paper, automorphisms of the algebra ofq-difference operators, as an associative algebra for arbitraryq and as a Lie algebra forq being not a root of unity, are determined. Project supported by the NNSF of China     

Inheritance of surjectivity for partial differential operators on spaces of real analytic functions

For an open set let A(Ω) be the space of real analytic functions on Ω. Improving our previous results, we prove a new quantitative characterization of the linear partial differential operators P(D) which are surjective on A(Ω). This implies that P(D) is surjective on if P(D) is surjective on A(Ω) for some Ω≠∅. Further inheritance properties for the surjectivity of P(D) on A(Ω) are also obtained.     

Representations of Lie algebras by non-skewselfadjoint operators in Hilbert space

We study non-skewselfadjoint representations of a finite dimensional real Lie algebra $\mathfrak{g}$. To this end we embed a non-skewselfadjoint representation of $\mathfrak{g}$ into a more complicated structure, that we call a $\mathfrak{g}$-operator vessel and that is associated to an overdetermined linear conservative input/state/output system on the corresponding simply connected Lie group $\mathfrak{G}$. We develop the frequency domain theory of the system in terms of representations of $\mathfrak{G}$, and introduce the joint characteristic function of a $\mathfrak{g}$-operator vessel which is the analogue of the classical notion of the characteristic function of a single non-selfadjoint operator. As the first non-commutative example, we apply the theory to the Lie algebra of the $ax+b$ group, the group of affine transformations of the line.