Convexity Inequalities for Positive Operators

We prove pointwise convexity (Jensen-type) inequalities of the form Open image in new window where F is a convex function defined on a convex subset of some Banach space X and T is the X-valued extension of a positive operator on some function space. Examples include the pointwise Hölder inequality T(fg) ≤ (Tf ^p )^{1/ p} (Tf ^q )^{1/ q} for a positive sublinear operator T. As applications we consider vector-valued conditional expectation and a ``real'' proof of the Riesz-Thorin theorem for positive operators.     

Invariant Subspaces of Positive Quasinilpotent Operators on Ordered Banach Spaces

In this paper we find invariant subspaces of certain positive quasinilpotent operators on Krein spaces and, more generally, on ordered Banach spaces with closed generating cones. In the later case, we use the method of minimal vectors. We present applications to Sobolev spaces, spaces of differentiable functions, and C*-algebras.      

Norm Inequalities for Sums of Positive Operators. II

It is shown that if A and B are positive operators on a separable complex Hilbert space, then for every unitarily invariant norm. When specialized to the usual operator norm ||·|| and the Schatten p-norms ||·||_p, this inequality asserts that and These inequalities improve upon some earlier related inequalities. Other norm inequalities for sums of positive operators are also obtained.     

Extremal Problems for Operators in Banach Spaces Arising in the Study of Linear Operator Pencils

Inspired by some problems on fractional linear transformations the authors introduce and study the class of operators satisfying the condition where stands for the spectral radius; and the class of Banach spaces in which all operators satisfy this condition, the authors call such spaces V-spaces. It is shown that many well-known reflexive spaces, in particular, such spaces as L_p(0,1) and C_p, are non-V-spaces if p 2; and that the spaces l_p are V-spaces if and only if 1 < p < . The authors pose and discuss some related open problems.     

Essential Spectral Inclusions for Operators on Banach Spaces

This paper centers on local spectral conditions that are both necessary and sufficient for the equality of the essential spectra of two bounded linear operators on complex Banach spaces that are intertwined by a pair of bounded linear mappings. In particular, if the operators T and S are intertwined by a pair of injective operators, then S is Fredholm provided that T is Fredholm and S has property (δ) in a neighborhood of 0. In this case, ind(T) ≤ ind(S), and equality holds precisely when the eigenvalues of the adjoint T* do not cluster at 0. By duality, we obtain refinements of results due to Putinar, Takahashi, and Yang concerning operators with Bishop’s property (β) intertwined by pairs of operators with dense range. Moreover, we establish an extension of a result due to Eschmeier that, under appropriate assumptions regarding the single-valued extension property, leads to necessary and sufficient conditions for quasi-similar operators to have equal essential spectra. In particular it turns out that the single-valued extension property plays an essential role in the preservation of the index in this context.      

Compact Differences of Weighted Composition Operators on Weighted Banach Spaces of Analytic Functions

Let be the weighted Banach space of analytic functions with a topology generated by weighted sup-norm. In the present article, we investigate the analytic mappings and which characterize the compactness of differences of two weighted composition operators on the space . As a consequence we characterize the compactness of differences of composition operators on weighted Bloch spaces.      

Isometric Equivalence of Certain Operators on Banach Spaces

A pair of operators on a Banach space X are isometrically equivalent if they are intertwined by a surjective isometry of X. We investigate the isometric equivalence problem for pairs of operators on specific types of Banach spaces. We study weighted shifts on symmetric sequence spaces, elementary operators acting on an ideal I of Hilbert space operators, and composition operators on the Bloch space. This last case requires an extension of known results about surjective isometries of the Bloch space.     

Triviality of the Peripheral Point Spectrum of Positive Semigroups on Atomic Banach Lattices

Generalizing a result of Keicher [4] we show that generators of positive C ₀-semigroups on super-atomic Banach lattices have trivial peripheral point spectrum provided they satisfy a certain growth condition.     

Differences of Composition Operators between Weighted Banach Spaces of Holomorphic Functions on the Unit Polydisk

We consider differences of composition operators between given weighted Banach spaces H ^∞ _v or H ⁰ _v of analytic functions defined on the unit polydisk D ^N with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. This paper is an extension of [6] where the one-dimensional case is treated. Received: May 15, 2007. Revised: October 8, 2007.     

Bivariate Function Spaces and the Embedding of Their Marginal Spaces

For a probability space we denote the marginal measures of , defined on Σ and Λ respectively, by and . If ρ is a function norm defined on marginal function norms ρ₁ and ρ₂ are defined on and . We find conditions which guarantee L_{ρ 1} + L_{ρ 2} to be embedded in L_ρ as a closed subspace. The problem is encountered in Statistics when estimating a bivariate distribution with known marginals. We find a condition which, applied to the binormal distribution in L², improves some known conditions.     

Inequalities for sums and direct sums of Hilbert space operators

We prove several singular value inequalities and norm inequalities involving sums and direct sums of Hilbert space operators. It is shown, among other inequalities, that if X and Y are compact operators, then the singular values of are dominated by those of X ⊕ Y. Applications of these inequalities are also given.     

On the spectral radius of positive operators on Banach sequence spaces

Let K₁,…,K_n be (infinite) non-negative matrices that define operators on a Banach sequence space. Given a function f:[0,∞)×…×[0,∞)→[0,∞) of n variables, we define a non-negative matrix and consider the inequality

     

On principal T-bands in a Banach lattice

It is shown that for every positive order continuous Riesz operatorT, defined on an order complete complex Banach latticeE which is separated by its Köthe dual, there exists a Frobenius decomposition ofE into a countable number of disjoint principalT-bands and a band on whichT is quasi-nilpotent. A basis for the generalized eigenspace ofT pertaining to its maximal eigenvalue is constructed and the positivity properties of its elements are studied. The distinguished eigenvalues ofT are characterized and it is also shown that the theory ofT-bands is symmetric with respect to the duality which exists betweenE and its Köthe dual. This generalizes aspects of work done by H.D. Victory and R.-J. Jang-Lewis.     

Positive Semigroups of Kernel Operators

Extending results of Davies and of Keicher on ℓ^p we show that the peripheral point spectrum of the generator of a positive bounded C₀-semigroup of kernel operators on L^p is reduced to 0. It is shown that this implies convergence to an equilibrium if the semigroup is also irreducible and the fixed space non-trivial. The results are applied to elliptic operators. Dedicated to the memory of H.H. Schaefer     

Discretizations of Integral Operators and Atomic Decompositions in Vector-Valued Weighted Bergman Spaces

We prove a general atomic decomposition theorem for weighted vector-valued Bergman spaces, which applies to duality problems and to the study of compact Toeplitz type operators.      

Wiener-Hopf Operators with Semi-almost Periodic Matrix Symbols on Weighted Lebesgue Spaces

Fredholm criteria and index formulas are established for Wiener-Hopf operators W(a) with semi-almost periodic matrix symbols a on weighted Lebesgue spaces where 1 < p < ∞, w belongs to a subclass of Muckenhoupt weights and . We also study the invertibility of Wiener-Hopf operators with almost periodic matrix symbols on . In the case N = 1 we also obtain a semi-Fredholm criterion for Wiener-Hopf operators with semi-almost periodic symbols and, for another subclass of weights, a Fredholm criterion for Wiener-Hopf operators with semi-periodic symbols. Work was supported by the SEP-CONACYT Project No. 25564 (México). The second author was also sponsored by the CONACYT scholarship No. 163480.     

Singular Integral Operators with Fixed Singularities on Weighted Lebesgue Spaces

The paper is devoted to study of singular integral operators with fixed singularities at endpoints of contours on weighted Lebesgue spaces with general Muckenhoupt weights. Compactness of certain integral operators with fixed singularities is established. The membership of singular integral operators with fixed singularities to Banach algebras of singular integral operators on weighted Lebesgue spaces with slowly oscillating Muckenhoupt weights is proved on the basis of Balakrishnans formula from the theory of strongly continuous semi-groups of closed linear operators. Symbol calculus for such operators, Fredholm criteria and index formulas are obtained.