Norm attaining operators and renormings of Banach spaces

We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc _o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+ε)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+ε)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.     

Closed linear operators between nonarchimedean Banach spaces

In this work we establish some basic properties of closed linear operators between nonarchimedean Banach spaces. As a consequence, we characterize the operators with closed range, and we establish the state diagram for closed linear operators when the underlying spaces satisfy a Hahn-Banach property.     

Nonlinear perturbations of linear accretive operators in Banach spaces

It is shown that a strongly continuous semi-group of nonlinear nonexpansive operators can be constructed as lim _n→∞ ((I+t/nB)⁻¹ (I+t/nB)⁻¹) ⁿ whereA is a linearm-accretive operator,B is a nonlinearm-accretive operator, andB satisfies a boundedness condition relative toA.     

Relative boundedness and relative compactness for linear operators in Banach spaces

If and are linear operators acting between Banach spaces, we show that compactness of relative to does not in general imply that has -bound zero. We do, however, give conditions under which the above implication is valid.

     

Hypercyclic operators failing the Hypercyclicity Criterion on classical Banach spaces

By a recent result of M. De La Rosa and C. Read, there exist hypercyclic Banach space operators which do not satisfy the Hypercyclicity Criterion. In the present paper, we prove that such operators can be constructed on a large class of Banach spaces, including or .     

Identification problem of two operators for nonlinear systems in Banach spaces

We consider the identification problem of two operators having different properties for the systems governed by nonlinear evolution equations. For the identification problem, we show the existence of optimal solutions and present necessary optimality conditions. We illustrate the approach on two examples.     

Adjoint for operators in Banach spaces

In this paper we show that a result of Gross and Kuelbs, used to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is used to extend well-known theorems of von Neumann and Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. This last result extends a theorem of Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on Banach spaces. As an application, we obtain a generalization of the Yosida approximator for semigroups of operators.

     

Automatic continuity of intertwining linear operators on Banach spaces

We extend the automatic continuity theory for linear operators θ:X→Y which intertwine two given bounded linear operatorsT∈L X andS∈L Y on Banach spacesX andY, respectively. This is done both by relaxing the intertwining conditionSθ=θT and by enlarging the classes of operatorsT, resp.S, well beyond the decomposable operators. Among the operatorsS captured by these extensions are multipliers on commutative semi-simple Banach algebras. dedicated to George Maltese on his 60th birthday     

Invariant Gaussian measures for operators on Banach spaces and linear dynamics

We give conditions for an operator T on a complex separableBanach space X with sufficiently many eigenvectors associatedto eigenvalues of modulus 1 to admit a non-degenerate invariantGaussian measure with respect to which it is weak-mixing. Theexistence of such a measure depends on the geometry of the Banachspace and on the possibility of parametrizing the -eigenvectorfields of T in a regular way. We also investigate the connectionwith frequent hypercyclicity.     

On Ulyanov inequalities in Banach spaces and semigroups of linear operators

Let X,Y be Banach spaces and {T(t):t≥0} be a consistent, equibounded semigroup of linear operators on X as well as on Y; it is assumed that {T(t)} satisfies a Nikolskii type inequality with respect to X and Y:T(2t)f_Y(t)T(t)f_X. Then an abstract Ulyanov type inequality is derived between the (modified) K-functionals with respect to (X,D_X((-A)^α)) and (Y,D_Y((-A)^α)),α>0, where A is the infinitesimal generator of {T(t)}. Particular choices of X,Y are Lorentz–Zygmund spaces, of {T(t)} are those connected with orthogonal expansions such as Fourier, spherical harmonics, Jacobi, Laguerre, Hermite series. Known characterizations of the K-functionals lead to concrete Ulyanov type inequalities. In particular, results of Ditzian and Tikhonov in the case , are partly covered.     

On perturbations of linear m-accretive operators on reflexive Banach spaces

In this note we prove some results on the m-accretivity of sums and products of linear operators. In particular we obtain the following theorem: LetA, B be two m-accretive operators on a reflexive Banach space. IfA is invertible and (A)^–1 B is accretive thenBA ^–1 andA+B are m-accretive.     

Contractibility of the linear group in Banach spaces of measurable operators

We show contractibility to a point of the linear group for a wide class of symmetric spaces of measurable operators affiliated with several concrete non-atomic semifinite von Neumann algebras.Research supported by the Australian Research Council     

Generic solutions for some perturbed optimization problem in non-reflexive Banach spaces

Let Z be a closed, boundedly relatively weakly compact, nonempty subset of a Banach space X, and J:Z→R a lower semicontinuous function bounded from below. If X₀ is a convex subset in X and X₀ has approximatively Z-property (K), then the set of all points x in X₀?Z for which there exists z₀∈Z such that J(z₀)+‖x−z₀‖=?(x) and every sequence {z_n}⊂Z satisfying lim_n→∞[J(z_n)+‖x−z_n‖]=?(x) for x contains a subsequence strongly convergent to an element of Z is a dense G_δ-subset of X₀?Z. Moreover, under the assumption that X₀ is approximatively Z-strictly convex, we show more, namely that the set of all points x in X₀?Z for which there exists a unique point z₀∈Z such that J(z₀)+‖x−z₀‖=?(x) and every sequence {z_n}⊂Z satisfying lim_n→∞[J(z_n)+‖x−z_n‖=?(x) for x converges strongly to z₀ is a dense G_δ-subset of X₀?Z. Here . These extend S. Cobzas's result [J. Math. Anal. Appl. 243 (2000) 344-356].     

Point initial value problem for linear functional differential equations in Banach spaces

Summary The problem of existence and uniqueness of solutions defined on the whole real line and satisfying given initial point data for general abstract linear functional differential equations is considered. The equation is not assumed to be of the delay type. The essence of the method presented here consists in the representation of a solution in the form analogous to the variation of constants formula known for linear ordinary differential equations. It is shown that such an approach can be effectively applied to the problem of existence and uniqueness of solutions satisfying an exponential growth estimate, provided that the deviation of the argument is sufficiently small. The proofs are based on the Banach fixed point principle. Detailed comparison and discussion of the hypotheses ensuring the existence and uniqueness of solutions are presented.