Search results for "Operator"
showing 10 items of 1427 documents
The Bishop–Phelps–Bollobás theorem for operators
2008
AbstractWe prove the Bishop–Phelps–Bollobás theorem for operators from an arbitrary Banach space X into a Banach space Y whenever the range space has property β of Lindenstrauss. We also characterize those Banach spaces Y for which the Bishop–Phelps–Bollobás theorem holds for operators from ℓ1 into Y. Several examples of classes of such spaces are provided. For instance, the Bishop–Phelps–Bollobás theorem holds when the range space is finite-dimensional, an L1(μ)-space for a σ-finite measure μ, a C(K)-space for a compact Hausdorff space K, or a uniformly convex Banach space.
*-Representations, seminorms and structure properties of normed quasi*-algebras
2008
The class of -representations of a normed quasi -algebra (X;A0) is in- vestigated, mainly for its relationship with the structure of (X;A0). The starting point of this analysis is the construction of GNS-like -representations of a quasi -algebra (X;A0) dened by invariant positive sesquilinear forms. The family of bounded invariant positive sesquilinear forms denes some seminorms (in some cases, C -seminorms) that provide useful information on the structure of (X;A0) and on the continuity properties of its -representations. 1. Introduction. A quasi -algebra is a couple (X;A0), where X is a vector space with involution , A0 is a -algebra and a vector subspace of X, and X is an A0-bimodule who…
Incomparable Banach spaces and operator semigroups
2002
Using the notions of total incomparability and total coincomparability of Banach spaces, we define two families of operator semigroups. We show that these semigroups are minimal, in the sense that they admit a perturbative characterization. Moreover, they allow us to characterize the corresponding incomparability classes.
Spectral invariance, ellipticity, and the Fredholm property for pseudodifferential operators on weighted Sobolev spaces
1992
The pseudodifferential operators with symbols in the Grushin classes \~S inf0 supρ,δ , 0 ≤ δ < ρ ≤ 1, of slowly varying symbols are shown to form spectrally invariant unital Frecher-*-algebras (Ψ*-algebras) in L(L 2(R n )) and in L(H γ st ) for weighted Sobolev spaces H infγ sup defined via a weight d function γ. In all cases, the Fredholm property of an operator can be characterized by uniform ellipticity of the symbol. This gives a converse to theorems of Grushin and Kumano-Ta-Taniguchi. Both, the spectrum and the Fredholm spectrum of an operator turn out to be independent of the choices of s, t and γ. The characterization of the Fredholm property by uniform ellipticity leads to an index …
Contractivity results in ordered spaces. Applications to relative operator bounds and projections with norm one
2016
This paper provides various “contractivity” results for linear operators of the form I−C where C are positive contractions on real ordered Banach spaces X. If A generates a positive contraction semigroup in Lebesgue spaces Lp(μ), we show (M. Pierre's result) that A(λ−A)−1 is a “contraction on the positive cone”, i.e. A(λ−A)−1x≤x for all x∈L+p(μ)(λ>0), provided that p⩾2. We show also that this result is not true for 1 ⩽ p<2. We give an extension of M. Pierre's result to general ordered Banach spaces X under a suitable uniform monotony assumption on the duality map on the positive cone X+. We deduce from this result that, in such spaces, I−C is a contraction on X+ for any positive projection…
Operators in Rigged Hilbert spaces: some spectral properties
2014
A notion of resolvent set for an operator acting in a rigged Hilbert space $\D \subset \H\subset \D^\times$ is proposed. This set depends on a family of intermediate locally convex spaces living between $\D$ and $\D^\times$, called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.
Maximal regularity for Kolmogorov operators in L2 spaces with respect to invariant measures
2006
Abstract We prove an optimal embedding result for the domains of Kolmogorov (or degenerate hypoelliptic Ornstein–Uhlenbeck) operators in L 2 spaces with respect to invariant measures. We use an interpolation method together with optimal L 2 estimates for the space derivatives of T ( t ) f near t = 0 , where T ( t ) is the Ornstein–Uhlenbeck semigroup and f is any function in L 2 .
On the Toeplitz algebras of right-angled and finite-type Artin groups
1999
The graph product of a family of groups lies somewhere between their direct and free products, with the graph determining which pairs of groups commute and which do not. We show that the graph product of quasi-lattice ordered groups is quasi-lattice ordered, and, when the underlying groups are amenable, that it satisfies Nica's amenability condition for quasi-lattice orders. As a consequence the Toeplitz algebras of these groups are universal for covariant isometric representations on Hilbert space, and their representations are faithful if the isometries satisfy a properness condition given by Laca and Raeburn. An application of this to right-angled Artin groups gives a uniqueness theorem …
Representable linear functionals on partial *-algebras
2012
A GNS-like *-representation of a partial *-algebra \({{\mathfrak A}}\) defined by certain representable linear functionals on \({{\mathfrak A}}\) is constructed. The study of the interplay with the GNS construction associated with invariant positive sesquilinear forms (ips) leads to the notions of pre-core and of singular form. It is shown that a positive sesquilinear form with pre-core always decomposes into the sum of an ips form and a singular one.
Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness
2012
Abstract We present some extension of a well-known fixed point theorem due to Burton and Kirk [T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 423–431] for the sum of two nonlinear operators one of them compact and the other one a strict contraction. The novelty of our results is that the involved operators need not to be weakly continuous. Finally, an example is given to illustrate our results.