Representation theorem for locally defined operators in the space of Whitney differentiable functions

Under the assumption that is perfect, a representation theorem for locally defined operators mapping the space C ^m (A) of Whitney differentiable functions into C ¹(A) is given and an open problem is presented.     

Continuity of the spectrum on a class of upper triangular operator matrices

Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A^○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A^○) for some operators X and B such that λ∉σp(B) and σ(A^○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP.     

Recovering a leading coefficient and a memory kernel in first-order integro-differential operator equations

We are concerned with the identification of the scalar functions a and k in the convolution first-order integro-differential equation u′(t)−a(t)Au(t)−k∗Bu(t)=f(t), 0?t?T, , in a Banach space X, where A and B are linear closed operators in X, A being the generator of an analytic semigroup of linear bounded operators. Taking advantage of two pieces of additional information, we can recover, under suitable assumptions and locally in time, both the unknown functions a and k. The results so obtained are applied to an n-dimensional integro-differential identification problem in a bounded domain in .     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

Retractions and contractibility in hyperspaces

Given a metric continuum X, let X² denote the hyperspace of all nonempty closed subsets of X. For each positive integer k let Ck(X) stand for the hyperspace of members of X² having at most k components. Consider mappings (where B∈Cm(X)) and both defined by A?A∪B. We give necessary and sufficient conditions under which these mappings are deformation retractions (under a special convention for φB). The conditions are related to the contractibility of the corresponding hyperspaces.     

Locally defined operators in the space of Whitney differentiable functions

For a given set A⊂Rⁿ$A\subset {\mathbb{R}}^n$ a representation theorem for locally defined operators mapping the space C^m(A)$C^m\left(A\right)$ of Whitney differentiable functions into C⁰(A)$C^0\left(A\right)$ is presented.     

Classification of contractively complemented Hilbertian operator spaces

We construct some separable infinite-dimensional homogeneous Hilbertian operator spaces and , which generalize the row and column spaces R and C (the case m=0). We show that a separable infinite-dimensional Hilbertian JC^∗-triple is completely isometric to one of , , , or the space Φ spanned by creation operators on the full anti-symmetric Fock space. In fact, we show that (respectively ) is completely isometric to the space of creation (respectively annihilation) operators on the m (respectively m+1) anti-symmetric tensors of the Hilbert space. Together with the finite-dimensional case studied in [M. Neal, B. Russo, Representation of contractively complemented Hilbertian operator spaces on the Fock space, Proc. Amer. Math. Soc. 134 (2006) 475-485], this gives a full operator space classification of all rank-one JC^∗-triples in terms of creation and annihilation operator spaces.We use the above structural result for Hilbertian JC^∗-triples to show that all contractive projections on a C^∗-algebra A with infinite-dimensional Hilbertian range are “expansions” (which we define precisely) of normal contractive projections from A** onto a Hilbertian space which is completely isometric to R, C, R∩C, or Φ. This generalizes the well-known result, first proved for B(H) by Robertson in [A.G. Robertson, Injective matricial Hilbert spaces, Math. Proc. Cambridge Philos. Soc. 110 (1991) 183-190], that all Hilbertian operator spaces that are completely contractively complemented in a C^∗-algebra are completely isometric to R or C. We use the above representation on the Fock space to compute various completely bounded Banach-Mazur distances between these spaces, or Φ.     

Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S^∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C⁻¹(I−λ₁A₁−?−λdAd)(λ₁B₁+?+λdBd) such that connecting operator has adjoint U^∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A₁,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A₁,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A₁,…,Ad satisfy an additional stability property.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

Construction of a certain circular unit and its applications

For an abelian number field k, let CS(k) be the group of circular units of k defined by Sinnott, and CW(k) be that suggested by Washington. In this paper, we construct an element in CW(k) for a real subfield k of conductor . We will see that the order of in the factor group CW(k)/CS(k) can be very large. As an application, we derive some information about the class number of k for special cases.     

Rationality theorems for Hecke operators on GLn

We define n families of Hecke operators for GL_n whose generating series are rational functions of the form q_k(u)⁻¹ where q_k is a polynomial of degree , and whose form is that of the kth exterior product. This work can be viewed as a refinement of work of Andrianov (Math. USSR Sb. 12(3) (1970)), in which he defined Hecke operators the sum of whose generating series was a rational function with nontrivial numerator and whose denominator was essentially .By a careful analysis of the Satake map which defines an isomorphism between a local Hecke algebra and a ring of symmetric polynomials, we define n families of (polynomial) Hecke operators and characterize their generating series as rational functions. We then give an explicit means by which to locally invert the Satake isomorphism, and show how to translate these polynomial operators back to the classical double coset setting. The classical Hecke operators have generating series of exactly the same form as their polynomial counterparts, and hence are of number-theoretic interest. We give explicit examples for GL₃ and GL₄.     

The crossed product by a partial endomorphism and the covariance algebra

Given a local homeomorphism where U⊆X is clopen and X is a compact and Hausdorff topological space, we obtain the possible transfer operators Lρ which may occur for given by α(f)=f○σ. We obtain examples of partial dynamical systems (XA,σA) such that the construction of the covariance algebra C^∗(XA,σA), proposed by B.K. Kwasniewski, and the crossed product by a partial endomorphism O(XA,α,L), recently introduced by the author and R. Exel, associated to this system are not equivalent, in the sense that there does not exist an invertible function ρ∈C(U) such that O(XA,α,Lρ)≅C^∗(XA,σA).     

Asplund sets, differentiability and subdifferentiability of functions in Banach spaces

We show that Asplund sets are effective tools to study differentiability of Lipschitz functions, and ε-subdifferentiability of lower semicontinuous functions on general Banach spaces. If a locally Lipschitz function defined on an Asplund generated space has a minimal Clarke subdifferential mapping, then it is TBY-uniformly strictly differentiable on a dense Gδ subset of X. Examples are given of locally Lipschitz functions that are TBY-uniformly strictly differentiable everywhere, but nowhere Fréchet differentiable.     

On the stable rank of algebras of operator fields over metric spaces

Let Γ be a finitely generated, torsion-free, two-step nilpotent group. Let C^*(Γ) denote the universal C^*-algebra of Γ. We show that , where for a unital C^*-algebra A, sr(A) is the stable rank of A, and where is the space of one-dimensional representations of Γ. In process, we give a stable rank estimate for maximal full algebras of operator fields over metric spaces.     

Cells in hyperspaces

Let C(X) denote the hyperspace of subcontinua of a continuum X. For A∈C(X), define the hyperspace . Let k∈N, k?2. We prove that A is contained in the core of a k-od if and only if C(A,X) contains a k-cell.     

Global minimum and orthogonality in C1-classes

In this paper we characterize the global minimum of an arbitrary function defined on a Banach space, in terms of a new concept of derivatives adapted for our case from a recent work due to D.J. Keckic (J. Operator Theory, submitted for publication). Using these results we establish several new characterizations of the global minimum of the map defined by F_ψ(X)=‖ψ(X)‖₁, where is a map defined by ψ(X)=S+φ(X) and φ:B(H)→B(H) is a linear map, S∈C₁, and . Further, we apply these results to characterize the operators which are orthogonal to the range of elementary operators.     

Asymptotic behavior of spectral functions for elliptic operators with non-smooth coefficients

We consider the asymptotic formula of spectral functions for elliptic operators with non-smooth coefficients of order 2m in . If the coefficients of top order are Hölder continuous of exponent τ∈(0,1], we can derive the remainder estimate of the form O(t^(n−θ)/2m) with any θ∈(0,τ). This result holds without the condition 2m>n, which was always assumed in many papers. We also show that the spectral function is differentiable up to order <m.     

K-group and similarity classification of operators

Let H be a complex separable Hilbert space and L(H) denote the collection of bounded linear operators on H. An operator A in L(H) is said to be a Cowen-Douglas operator if there exist Ω, a connected open subset of complex plane C, and n, a positive integer, such that

(a)

(b)

  for z in Ω;

(c)

; and

(d)

for z in Ω.

In the paper, we give a similarity classification of Cowen-Douglas operators by using the ordered K-group of the commutant algebra as an invariant, and characterize the maximal ideals of the commutant algebras of Cowen-Douglas operators. The theorem greatly generalizes the main result in (Canada J. Math. 156(4) (2004) 742) by simply removing the restriction of strong irreducibility of the operators. The research is also partially inspired by the recent classification theory of simple AH algebras of Elliott-Gong in (Documenta Math. 7 (2002) 255; On the classification of simple inductive limit C^*-algebras, II: The isomorphism theorem, preprint.) (also see (J. Funct. Anal. (1998) 1; Ann. Math. 144 (1996) 497; Amer. J. Math. (1996) 187)).     

Integral of the Clarke subdifferential mapping and a generalized Newton-Leibniz formula

In the theory of Lebesgue integration it has been proved that if f is a real Lipschitz function defined on a segment [a,b]⊂R, then the Newton-Leibniz formula (the fundamental theorem of calculus) holds. This paper extends the fact to the case where the Fréchet derivative f^′(⋅) (which is defined almost everywhere on [a,b] by the Rademacher theorem) and the Lebesgue integral are replaced, respectively, by the Clarke subdifferential mapping ∂_Cf(⋅) and the Aumann (set-valued) integral. Among other things, we show that and the equality is valid if and only if f is strictly Hadamard differentiable almost everywhere on [a,b]. The result is derived from a general representation formula, which we obtain herein for the integral of the Clarke subdifferential mapping of a Lipschitz function defined on a separable Banach space.