Overiterated Linear Operators and Asymptotic Behaviour of Semigroups

The results provided hereby are related to the asymptotic behaviour of certain strongly continuous semigroups, which may be expressed in terms of iterates of positive linear operators, in the sense of Altomare’s theory. We present some applications to concrete cases involving continuous and discrete type operators, namely the Beta and the Stancu operators.      

A Class of Integral Operators on the Unit Ball of \mathbb{C}^{n}

For real parameters a, b, c, and t, where c is not a nonpositive integer, we determine exactly when the integral operator

On The Extended Eigenvalues of Some Volterra Operators

We show that a large class of compact quasinilpotent operators has extended eigenvalues. As a consequence, if V is such an operator, then the associated spectral algebra contains its commutant {V}' as a proper subalgebra.     

Elements of the Theory of Linear Volterra Operators in Banach Spaces

The new definition of Volterra operator introduced in [5] allows specification of the classical theory of linear equations in Banach spaces to equations with such operators. Here we specially address relations between properties of the given linear equation with Volterra operator and properties of its conjugate. As well we treat the theory of Noetherian and Fredholm equations.     

On the Cowen-Douglas Class for Banach Space Operators

In [3], M. J. Cowen and R. G. Douglas prove that the adjoint of a Hilbert space operator T is in the class if and only if T is unitarily equivalent with the operator M _z on a Hilbert space -valued analytic functions, where M _z denotes the operator of multiplication by the independent variable. The proof involves holomorphic vector bundles and Grauert’s theorem. In this paper we use a theorem by I. Gohberg and L. Rodman [4] to give a more elementary proof of this fact, which also works for Banach space operators.      

Examples of operators on <Emphasis Type="Italic">C</Emphasis>[0, 1] distinguishing certain operator ideals

We prove that for some classes of weakly compact operators on the space C[0, 1] the conditions that they are compact, absolutely summing and nuclear are distinct and connected to vector-valued sequence spaces. Received: 22 February 2006     

Test Function Criteria for Hankel Operators

It is well known that there are classes of test functions such that a Hankel operator is bounded if and only if it is bounded on those functions. Criteria are derived which determine whether a Hankel operator is compact or belongs to a particular Schatten class, in terms of its action on those test functions.     

Hyperinvariant Subspace Problem for Quasinilpotent Operators

In this paper we consider the hyperinvariant subspace problem for quasinilpotent operators. Let denote the class of quasinilpotent quasiaffinities Q in such that Q ^* Q has an infinite dimensional reducing subspace M with Q ^* Q| _M compact. It was known that if every quasinilpotent operator in has a nontrivial hyperinvariant subspace, then every quasinilpotent operator has a nontrivial hyperinvariant subspace. Thus it suffices to solve the hyperinvariant subspace problem for elements in . The purpose of this paper is to provide sufficient conditions for elements in to have nontrivial hyperinvariant subspaces. We also introduce the notion of “stability” of extremal vectors to give partial solutions to the hyperinvariant subspace problem.      

Finite Sections of Band-Dominated Operators: l p -Theory

We propose an algebraic approach to the stability problem for the finite sections of general band-dominated operators acting on for 1 < p < ∞. This approach allows us to get new results which previously were known mainly for p = 2. One of the main results shows that a band-dominated operator is Fredholm if and only if the approximation numbers of its finite sections have a special behavior. Submitted: June 14, 2007. Accepted: November 5, 2007.     

Norm Attaining Operators and Pseudospectrum

It is shown that if 1 < p < ∞ and X is a subspace or a quotient of an ℓ_p-direct sum of finite dimensional Banach spaces, then for any compact operator T on X such that ∥I + T∥ > 1, the operator I + T attains its norm. A reflexive Banach space X and a bounded rank one operator T on X are constructed such that ∥I + T∥  > 1 and I + T does not attain its norm. The author would like to thank E. Shargorodsky for his interest and comments.     

Xia Spectrum for Some Class of Operators

In this paper, we introduce Xia spectra of n-tuples of operators satisfying |T ²| ≥ U|T ²|U* for the polar decomposition of T = U|T| and we extend Putnam’s inequality to these tuples [7]. This research is partially supported by Grant-in-Aid Research No.17540176.     

A Remark on Support of the Principal Function for Class A Operators

Let be an invertible class A operator such that . Then we show that , where gT is the principal function of T. Moreover, we show that if T is pure, then .     

Positive Schatten(-Herz) Class Toeplitz Operators on Pluriharmonic Bergman Spaces

We study characterizations of arbitrary positive Toeplitz operators of Schatten (or Schatten-Herz) type in terms of averaging functions and Berezin transforms of symbol functions on the ball of pluriharmonic Bergman space. This work was supported by a Hanshin University Research Grant.     

The Fredholm Index of Locally Compact Band-dominated Operators on L^p \,(\user2{\mathbb{R}})

We establish a necessary and sufficient criterion for the Fredholmness of a general locally compact band-dominated operator A on and solve the long-standing problem of computing its Fredholm index in terms of the limit operators of A. The results are applied to operators of convolution type with almost periodic symbol.     

Hyponormality of Toeplitz Operators with Polynomial and Symmetric-Type Symbols

Suppose that is a trigonometric polynomial of the form (z) = ^N_n=-N a_n zⁿ. It is well-known that T is normal if and only if | a_{– N}| =  | a_N| and the Fourier coefficients of satisfy the following symmetry condition:

On the Structure of L 1 of a Vector Measure via its Integration Operator

Geometric and summability properties of the integration operator associated to a vector measure m can be translated in terms of structure properties of the space L¹(m). In this paper we study the cases of the integration operator being: (i) p-concave on L^p(m), or (ii) positive p-summing on L¹(m) (where ). We prove that (i) is equivalent to saying that L¹(m) contains continuously the L^p space of a (non-negative scalar) control measure for m. On the other hand, we show that (ii) holds if and only if L¹(m) is order isomorphic to the L¹ space of a non-negative scalar measure. J.M. Calabuig was supported by MEC and FEDER (MTM2005-08350-C03-03) and Generalitat Valenciana (GV/2007/191). J. Rodríguez was supported by MEC and FEDER (MTM2005-08379) and Generalitat Valenciana (GVPRE/2008/312). E.A. Sánchez-Pérez was supported by MEC and FEDER (MTM2006-11690-C02-01).     

Operator Valued Functions with Pick Operators Having Negative Subspaces of Bounded Dimensions

Functions whose values are bounded linear Hilbert space operators (each operator may be defined on its own subspace of the ambient Hilbert space), the domain of definition is contained in the open unit disc, and having the following property κ, are studied. (κ): All Pick operators associated with the function have the dimensions of their spectral subspace corresponding to the negative part of the spectrum bounded above by a fixed nonnegative integer κ, and the bound κ is attained. No a priori hypotheses concerning regularity of the functions are assumed. A particular class of functions, called standard functions, is introduced, and the corresponding nonnegative integer κ is identified for standard functions. It is proved that every function with property (κ) can be extended to a standard function with property (κ), for the same κ. This result is interpreted as a result on interpolation. As an application, maximal (with respect to the extension relation) functions with the property κ, for a fixed κ, are studied in terms of standard functions. Received: August 5, 2007., Accepted: October 24, 2007.     

Hilbert-Schmidt Hankel Operators on the Bergman Space of Planar Domains

In this paper we study the problem of the membership of H _ϕ in the Hilbert-Schmidt class, when and Ω is a planar domain. We find a necessary and sufficient condition.We apply this result to the problem of joint membership of H _φ and in the Hilbert-Schmidt class. Using the notion of Berezin Transform and a result of K. Zhu we are able to give a necessary and sufficient condition. Finally, we recover a result of Arazy, Fisher and Peetre on the case with φ holomorphic.     

On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D

Let D be a quaternion division algebra with an involution of the second kind. We show that the quotient group SU(1,D)/[U(1,D),U(1,D)] is nontrivial in general. For global fields, we completely determine this group. Received: 11 May 2007, Revised: 15 December 2007     

Ideal-Triangularizability of Semigroups of Positive Operators

Let be a multiplicative semigroup of positive operators on a Banach lattice E such that every is ideal-triangularizable, i.e., there is a maximal chain of closed subspaces of E that consists of closed ideals invariant under S. We consider the question under which conditions the whole semigroup is simultaneously ideal-triangularizable. In particular, we extend a recent result of G. MacDonald and H. Radjavi. We also introduce a class of positive operators that contains all positive abstract integral operators when E is Dedekind complete.