Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ₀ and is boundedly invertible in the endpoints of Δ₀, then the ‘embedding’ of the original Hilbert space into the Hilbert space , where the linearization of A(z) acts, is in fact an isomorphism between a subspace of and . As a consequence, properties of the local spectral function of A(z) on Δ₀ and a so-called inner linearization of the operator function A(z) in the subspace are established.      

Algebras Generated by the Bergman and Anti-Bergman Projections and by Multiplications by Piecewise Continuous Functions

The C*-algebra generated by the Bergman and anti-Bergman projections and by the operators of multiplication by piecewise continuous functions on the Lebesgue space L²(Π) over the upper half-plane is studied. Making use of a local principle, limit operators techniques, and the Plamenevsky results on two-dimensional singular integral operators with coefficients admitting homogeneous discontinuities we reduce the study to simpler C*-algebras associated with points and pairs We construct a symbol calculus for unital C*-algebras generated by n orthogonal projections sum of which equals the unit and by m one-dimensional orthogonal projections. Such algebras are models of local algebras at points z ∈∂Π being the discontinuity points of coefficients. A symbol calculus for the C*- algebra and a Fredholm criterion for the operators are obtained. Finally, a C*-algebra isomorphism between the quotient algebra where is the ideal of compact operators, and its analogue for the unit disk is constructed.     

The Essential Norm of Hankel Operators on the Weighted Bergman Spaces with Exponential Type Weights

Let denote the closed subspace of consisting of analytic functions in the unit disc . For certain class of subharmonic functions and , it is shown that the essential norm of Hankel operator is comparable to the distance norm from H_f to compact Hankel operators.     

Operator Matrices: SVEP and Weyl’s Theorem

It is known that if and are Banach space operators with the single-valued extension property, SVEP, then the matrix operator has SVEP for every operator and hence obeys Browder’s theorem. This paper considers conditions on operators A, B, and M₀ ensuring Weyls theorem for operators M_C.     

Ergodic Properties for Regular A-Contractions

The current article pleads for the possibility to obtain an orthogonal decomposition of a Hilbert space which is induced by a regular A-contraction defined in [9, 10], A being a positive operator on . The decomposition generalizes the well-known decomposition related to a contraction T of , which gives the ergodic character of T. This decomposition is being used to prove certain versions for regular A-contractions of the mean ergodic theorem, as well as a version of Patil’s theorem from [8]. Also, we characterize the solutions of corresponding functional equations in the range of A^1/2, by analogy with the result of Lin-Sine in [7].     

Non- (Quantum) Differentiable C 1-Functions in the Spaces with Trivial Boyd Indices

If E is a separable symmetric sequence space with trivial Boyd indices and is the corresponding ideal of compact operators, then there exists a C¹-function f_E, a self-adjoint element and a densely defined closed symmetric derivation δ on such that , but      

Sampling Theory Associated with a Symmetric Operator with Compact Resolvent and De Branges Spaces

Let be a symmetric operator with compact resolvent defined in a Hilbert space For any fixed we consider an entire function K_a which involves the resolvent of Associated with K_a we obtain, by duality in a Hilbert space of entire functions which becomes a De Branges space of entire functions. This property provides a characterization of regardless of the anti-linear mapping which has as its range space. There exists also a sampling formula allowing to recover any function in from its samples at the sequence of eigenvalues of This work has been supported by the grant BFM2003–01034 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.     

Lie and Jordan Ideals in Reflexive Algebras

A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper, we show, for a weakly closed linear subspace of a CDCSL algebra , that is a Lie ideal if and only if for all invertibles A in , and that is a Jordan ideal if and only if it is an associative ideal.     

Additive maps preserving the minimum and surjectivity moduli of operators

Let be a complex Hilbert space and let be the algebra of all bounded linear operators on . We characterize additive maps from onto itself preserving different spectral quantities such as the minimum modulus, the surjectivity modulus, and the maximum modulus of operators. Received: 15 July 2008     

Symmetrization of Closure Operators and Visibility

For an arbitrary set E and a given closure operator , we want to construct a symmetric closure operator via some – possibly infinite – iteration process. If E is finite, the corresponding symmetric closure operator . defines a matroid. If and is the convex closure operator, turns out to be the affine closure operator. Moreover, we apply the symmetrization process to closure operators induced by visibility. Received March 9, 2005     

Hausdorff dimension of limit sets of discrete subgroups of higher rank Lie groups

Let X be a globally symmetric space of noncompact type, and a discrete subgroup. Introducing an appropriate notion of Hausdorff measure on the geometric boundary of , we prove that for regular boundary points , the Hausdorff dimension of the radial limit set in is bounded above by the exponential growth rate of the number of orbit points close in direction to . Furthermore, for Zariski dense discrete groups we construct -invariant densities with support in every G-invariant subset of the limit set and study their properties. For a class of groups which generalises convex cocompact groups in the rank one setting, these densities allow to give a sharp estimate on the Hausdorff dimension of the radial limit set in each subset .     

Algebraic Properties of Toeplitz Operators with Radial Symbols on the Bergman Space of the Unit Ball

In this paper, we discuss some algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball in . We first determine when the product of two Toeplitz operators with radial symbols is a Toeplitz operator. Next, we investigate the zero-product problem for several Toeplitz operators with radial symbols. Also, the corresponding commuting problem of Toeplitz operators whose symbols are of the form is studied, where and φ is a radial function. Ze-Hua Zhou: supported in part by the National Natural Science Foundation of China (Grand Nos.10671141, 10371091).     

Sums of Bisectorial Operators and Applications

We study sums of bisectorial operators on a Banach space X and show that interpolation spaces between X and D(A) (resp. D(B)) are maximal regularity spaces for the problem Ay + By = x in X. This is applied to the study of regularity properties of the evolution equation u′ + Au = f on for or and the evolution equation u′ + Au = f on [0, 2π] with periodic boundary condition u(0) = u(2π) in or      

On a Poincaré-type Inequality for Energy Forms in L p

We consider Dirichlet spaces ( ) in L ² and more general energy forms in L ^p , . For the latter we introduce the notions of an extended ’Dirichlet’ space and a transient form. Under the assumption that , resp. , are compactly embedded in L ², resp. L ^p , we prove a Poincaré inequality for transient (Dirichlet) forms. If both and its adjoint are sub-Markovian semigroups, we show that the transience of T _t is independent of ) and that it is implied by the transience of the energy form of and the form belonging to .     

On the Range of the Aluthge Transform

Let be the algebra of all bounded linear operators on a complex separable Hilbert space For an operator let be the Aluthge transform of T and we define for all where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range of Δ. We prove that R(Δ) is neither closed nor dense in However R(Δ) is strongly dense if is infinite dimensional. An erratum to this article is available at .     

A band limited and Besov class functional calculus for Tadmor-Ritt operators

For Banach space operators T satisfying the Tadmor-Ritt condition a band limited H^∞ calculus is established, where and a is at most of the order C(T)⁵. It follows that such a T allows a bounded Besov algebra B_{∞ 1}⁰ functional calculus, These estimates are sharp in a convenient sense. Relevant embedding theorems for B_{∞ 1}⁰ are derived. Received: 25 October 2004; revised: 31 January 2005     

An Operator Transform from Class A to the Class of Hyponormal Operators and its Application

In this paper, we shall give an operator transform from class A to the class of hyponormal operators. Then we shall show that and in case T belongs to class A. Next, as an application of we will show that every class A operator has SVEP and property (β).     

Equations arising from Jordan *-derivation pairs

Summary. We study certain functional equations derived from the definition of a Jordan *-derivation pair. More precisely, if A is a complex *-algebra and M is a bimodule over A, having the structure of a complex vector space compatible with the structure of A, such that implies m = 0 and implies m = 0 and if are unknown additive mappings satisfying then E and F can be represented by double centralizers. The obtained result implies that one of the equations in the definition of a Jordan *-derivation pair is redundant. Furthermore, a remark on the extension of this result to unknown additive mappings such that is given in a special case.     

Commutators on Half-Spaces

We study the boundedness and compactness of commutators on , where and are defined by and respectively. If satisfies some upper and lower estimates, then we obtain a necessary and sufficient condition for to be bounded or compact on for . The reproducing kernel of the harmonic Bergman space of can be shown to satisfy all the required estimates. Our result is the real variable analogue of the complex variable one for commutators associated with an analytic reproducing kernel.     

On the Root Functions of General Elliptic Boundary Value Problems

We consider a boundary value problem for an elliptic differential operator of order 2m in a domain . The boundary of is smooth outside a smooth manifold Y of dimension 0 ≤ q < n − 1, and bears edge type singularities along Y . The Lopatinskii condition is assumed to be fulfilled on the smooth part of . The corresponding spaces are weighted Sobolev spaces , and this allows one to define ellipticity of weight γ for the problem. The resolvent of the problem is assumed to possess rays of minimal growth. The main result says that if there are rays of minimal growth with angles between neighbouring rays not exceeding π(γ + 2m)/n, then the root functions of the problem are complete in . In the case of second order elliptic equations the results remain true for all domains with Lipschitz boundary. Communicated by Michael Shapiro. Submitted: May 24, 2006; Accepted: June 15, 2006