Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces

In this paper, the class of nonspreading mappings in Banach spaces is introduced. This class contains the recently introduced class of firmly nonexpansive type mappings in Banach spaces and the class of firmly nonexpansive mappings in Hilbert spaces. Among other things, we obtain a fixed point theorem for a single nonspreading mapping in Banach spaces. Using this result, we also obtain a common fixed point theorem for a commutative family of nonspreading mappings in Banach spaces. Received: 10 August 2007     

Point localization and spectral theory for symmetric random operators

In a Hilbert space context, we propose a rather general notion of “random operators” which allows for taking stochastic limits. After establishing a connection with measurable fields of closed operators, we may speak of a spectral theory for symmetric random operators. Received: 18 December 2008     

Remarks on u 0-positive operators

We show that the concept of a linear operator being u₀-positive, and refinements of some classical results of Krasnosel’skiĭ, can be used, in conjunction with the theory of fixed point index, to give some short proofs of existence and nonexistence of positive solutions for nonlinear maps. Dedicated to Felix Browder on the occasion of his 80th birthday     

On the spectrum of multivariable weighted composition operators

In this paper we study the point spectrum of the operator

Perturbations of differentiable semigroups

If (A, D(A)) generates a C ₀-semigroup T on a Banach space X and then (A + B, D(A)) is also the generator of a C ₀-semigroup, S _B . There are easy examples to show that if T is eventually differentiable then S _B need not be eventually differentiable. In 1995 an example was constructed to show that if T is immediately differentiable then S _B need not be immediately differentiable. In this paper we establish necessary and sufficient conditions on the generator (A, D(A)) of T which ensure that eventual or immediate differentiability of T is inherited by S _B for all . We are therefore able to give a characterization of the immediately and eventually differentiable C ₀-semigroups for which differentiability is a stable property under bounded perturbations of the generator. We also prove a characterization of the C ₀-semigroups for which the norm of the resolvent of the generator decays on vertical lines and a new characterization of the Crandall-Pazy class of semigroups. We are grateful to Charles Batty and Tom Ransford for helpful discussions and to the referee for their constructive comments.     

On the nuclearity of integral operators

Let X be a nonempty measurable subset of and consider the restriction of the usual Lebesgue measure σ of to X. Under the assumption that the intersection of X with every open ball of has positive measure, we find necessary and sufficient conditions on a L²(X)-positive definite kernel in order that the associated integral operator be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed when K is just an element of .      

Bicommutants of Toeplitz operators

In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant. Received: 11 March 2008     

A generalized flat extension theorem for moment matrices

In this note we prove a generalization of the flat extension theorem of Curto and Fialkow (Memoirs of the American Mathematical Society, vol. 119. American Mathematical Society, Providence, 1996) for truncated moment matrices. It applies to moment matrices indexed by an arbitrary set of monomials and its border, assuming that this set is connected to 1. When formulated in a basis-free setting, this gives an equivalent result for truncated Hankel operators.      

Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces

By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space on the unit disk can be represented as where Φ is an inner multiplier of . This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert space will be called Beurling decomposable if there exist (operator-valued) multipliers such that and . We characterize the finite-codimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules , every finite-codimensional submodule M can be written as with suitable polynomials p _i .      

Applications of a theorem of Singerman about Fuchsian groups

Assume that we have a (compact) Riemann surface S, of genus greater than 2, with , where is the complex unit disc and Γ is a surface Fuchsian group. Let us further consider that S has an automorphism group G in such a way that the orbifold S/G is isomorphic to where is a Fuchsian group such that and has signature σ appearing in the list of non-finitely maximal signatures of Fuchsian groups of Theorems 1 and 2 in [6]. We establish an algebraic condition for G such that if G satisfies such a condition then the group of automorphisms of S is strictly greater than G, i.e., the surface S is more symmetric that we are supposing. In these cases, we establish analytic information on S from topological and algebraic conditions. Received: 4 April 2008     

Mean ergodicity of multiplication operators in weighted spaces of holomorphic functions

Multiplication operators in weighted Banach (and locally convex) spaces of functions holomorphic in the unit disc are well known. In this note we investigate the connection between power boundedness, mean ergodicity and uniform mean ergodicity of such operators. Received: 13 October 2008, Revised: 18 November 2008     

Existence and extensions of positive linear operators

In this paper we develop some unified methods, based on the technique of the auxiliary sublinear operator, for obtaining extensions of positive linear operators. In the first part, a version of the Mazur-Orlicz theorem for ordered vector spaces is presented and then this theorem is used in diverse applications: decomposition theorems for operators and functionals, minimax theory and extensions of positive linear operators. In the second part, we give a general sufficient condition (an implication between two inequalities) for the existence of a monotone sublinear operator and of a positive linear operator. Some particular cases in which this condition becomes necessary are also studied. Dedicated to Prof. Romulus Cristescu on his 80th birthday     

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.     

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     

Norm Estimates of the Partial Transpose Map on the Tensor Products of Matrices

We present a norm estimate for the partial transpose map Θ on the tensor product

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.     

Banach algebra techniques for spectral multiplicity

In this paper, we use Banach algebra techniques to prove addition formulas for spectral multiplicities of direct sums of operator families. This work has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) with project 107T649.     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Estimates of holomorphic functions in zero-free domains

We study functions f(z) holomorphic in having the property f(z) ≠ 0 for 0 < Im z < 1 and we obtain lower bounds for |f(z)| for 0 < Im z < 1. In our analysis we deal with scalar functions f(z) as well as with operator valued holomorphic functions I + A(z) assuming that A(z) is a trace class operator for and I + A(z) is invertible for 0 < Im z < 1 and is unitary for . A. Borichev was partially supported by the ANR project DYNOP.     

Perturbation theorems for holomorphic semigroups

The concept of the gap function is used to give new perturbation results for generators of holomorphic semigroups. In particular, we show that if A is the generator of a holomorphic semigroup on a Banach space and , then every closed linear operator C such that for some and