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 .      

Estimates on the Green’s Function and Existence of Positive Solutions of Nonlinear Singular Elliptic Equations in the Half Space

We establish a new 3G-Theorem for the Green’s function for the half space We exploit this result to introduce a new class of potentials that we characterize by means of the Gauss semigroup on . Next, we define a subclass of and we study it. In particular, we prove that properly contains the classical Kato class . Finally, we study the existence of positive continuous solutions in of the following nonlinear elliptic problem

Approximation of p-multiplier Operators via their Spectral Projections

Conditions for a p-multiplier $\psi: {\mathbb{Z}} \to {\mathbb{C}}Conditions for a p-multiplier are presented which ensure that the corresponding operator T_ψ, acting in , can be approximated by linear combinations of p-multiplier projections coming from the uniform operator closed, unital algebra of operators generated by T_ψ. Functions of bounded variation on play an important role, as do certain Λ (p)-sets. Dedicated to the memory of H. H. Schaefer Werner J. Ricker: Former Alexander von Humboldt Fellow at the Universit?t Tübingen, hosted by Prof. H.H. Schaefer from Sept. 1987 – Feb. 1988.     

Carleson Measures for Spaces of Dirichlet Type

If 0 < p < ∞ and α > − 1, the space consists of those functions f which are analytic in the unit disc and have the property that f ′ belongs to the weighted Bergman space A_α^p. In 1999, Z. Wu obtained a characterization of the Carleson measures for the spaces for certain values of p and α. In particular, he proved that, for 0 < p ≤ 2, the Carleson measures for the space are precisely the classical Carleson measures. Wu also conjectured that this result remains true for 2 < p < ∞. In this paper we prove that this conjecture is false. Indeed, we prove that if 2 < p < ∞, then there exists g analytic in such that the measure μ_g,p on defined by dμ_g,p (z) = (1 − |z|²)^{p - 1}| g ′ (z)|^p dx dy is not a Carleson measure for but is a classical Carleson measure. We obtain also some sufficient conditions for multipliers of the spaces      

Multivariable moment problems

In this paper we solve moment problems for Poisson transforms and, more generally, for completely positive linear maps on unital C^*-algebras generated by universal row contractions associated with , the free semigroup with n generators. This class of C*-algebras includes the Cuntz-Toeplitz algebra (resp. ) generated by the creation operators on the full (resp. symmetric, or anti-symmetric)) Fock space with n generators. As consequences, we obtain characterizations for the orbits of contractive Hilbert modules over complex free semigroup algebras such as ,and, more generally, the quotient algebra , where J is an arbitrary two-sided ideal of . All these results are extended to the generalized Cuntz algebra , where G_i⁺ are the positive cones ofdiscrete subgroups G_i⁺ of the real line . Moreover, we characterize the orbits of Hilbert modules over the quotient algebra , where J is an arbitrary two-sided ideal ofthe free semigroup algebra .     

Gelfand-Hille type theorems in ordered Banach algebras

We consider the Gelfand-Hille Theorems, specifically conditions under which an element in an ordered Banach algebra (A,C) with spectrum {1} is the identity of the algebra. In particular we show that for , where C is a closed normal algebra cone, if and x is doubly Abel bounded then x = 1. Furthermore in the case where and C is a closed proper algebra cone, then x = 1 if and only if x^L is Abel bounded and for some .      

Compactness of Bochner representable operators on Orlicz spaces

We study compactness properties of linear operators from an Orlicz space L^Φ provided with a natural mixed topology to a Banach space (X, || · ||_X). We derive that every Bochner representable operator is -compact. In particular, it is shown that every Bochner representable operator is (τ(L^∞, L¹), || · ||_X)-compact.      

Subharmonic Functions in the Unit Ball

For functions u subharmonic in the unit ball B_N of , this paper compares the growth of the repartition function of their Riesz measure μ with the growth of u near the boundary of B_N. Cases under study are: and , with A, B, γ positive constants and if N=2 or if N≥ 3. This paper contains several integral results, as for instance: when ∫_BN u⁺(x)[-ω^′(|x|²)]dx < +∞ for some positive decreasing C¹ function ω, it is proved that .     

Regular orbits and positive directions

Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14] by F. León and V. Müller. In this paper we study positive supercyclicity for operators A of the form , with , defined on . We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central throughout the paper.      

<Emphasis Type="Italic">R</Emphasis>-bounded Representations of <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> (<Emphasis Type="Italic">G</Emphasis>)

We investigate R-bounded representations , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism , we are then able to analyze certain classical homomorphisms U (e.g. translations in L^p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators. Dedicated to the memory of H. H. Schaefer     

Spectral Exponent of Finite Sums of Weighted Positive Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Spaces

In this paper we investigate the spectral exponent, i.e. logarithm of the spectral radius of operators having the form

Characterization of nearly Schroeder-Bernstein quadruples for Banach spaces

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y . Let (p, q, r, s) be a quadruple in with p + q  ≥  2 and r + s ≥  2. Suppose that for every pair of Banach spaces X and Y isomorphic to complemented subspaces of each other and satisfying the following Decomposition Scheme

A characterization of complex lattice homomorphisms

In this paper we discuss the problem of how to recognize a complex lattice homomorphism on the complexification of a real vector lattice L from its behavior on a small subset of .      

Herz-type Triebel-Lizorkin Spaces, Ⅰ

Let s ∈R,0〈β≤∞, 0〈 q, p〈 ∞ and-n/q〈α. In this paper the authors introduce the Herz-type Triebel-Lizorkin spaces,Kq^α,pFβ^s（R^n）andKq^α,pFβ^s（R^n）which are the generalizations of the well-known Herz-type spaces and the inhomogeneous Triebel-Lizorkin spaces, Some properties on these Herz-type Triebel Lizorkin spaces are also given.     

Some properties of essential spectra of a positive operator, II

Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σ_ew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σ_ef(T), then for every a ≠ 0, where T₋₁ is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if , where , then an operator on D_i is band irreducible.      

Herz Classes and Toeplitz Operators in the Disk

In this paper we decompose into diadic annuli and consider the class S_p,q of Toeplitz operators T_φ for which the sequence of Schatten norms belongs to ℓ^q, where φ_n = φχ A_n. We study the boundedness and compactness of the operators in S_p,q and we describe the operators T_φ , φ ≥ 0 in these spaces in terms of weighted Herz norms of the averaging operator of the symbols φ.     

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.     

The Schur–Potapov Algorithm for Sequences of Complex p × q Matrices. I

The main theme of this paper is to characterize distinguished subclasses of the matricial Schur class in terms of Taylor coefficients. Starting point of our investigations is the observation that the Taylor coefficient sequences of functions from are exactly the infinite p  ×  q Schur sequences. We draw our attention mainly to the subclass of which consists of all p ×  q Schur functions for which the corresponding Taylor coefficient sequences are nondegenerate p  ×  q Schur sequences. Using an appropriate adaptation of the Schur–Potapov algorithm for functions belonging to to infinite sequences of complex p  ×  q matrices we obtain an one-to-one correspondence between infinite nondegenerate p  ×  q Schur sequences and the set of all infinite sequences (E_j)_j=0^∞ of strictly contractive complex p  ×  q matrices. Taking into account the construction of this gives us an one-to-one correspondence between and the set of all infinite sequences (E_j)_j=0^∞ of strictly contractive complex p  ×  q matrices. Hereby, (E_j)_{j =0}^∞ is called the sequence of Schur–Potapov parameters (shortly SP-parameters) of f. Communicated by Daniel Alpay. Submitted: August 17, 2006; Accepted: September 13, 2006     

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

Spaces of Operators,the ψ-Daugavet Property,and Numerical Indices

Suppose ψ : [0, ∞) → [1, ∞) is a strictly increasing function. A Banach space X is said to have the ψ-Daugavet Property if the inequality holds for every compact operator T : X → X. We show that, if 1 < p < ∞ and K(ℓ_p)↪ X ↪ B(ℓ_p), then X has the ψ-Daugavet Property with (here and c_p is an absolute constant). We also prove that a C^*-algebra A is commutative if and only if for any . Together, these results allow us to distinguish between some types of von Neumann algebras by considering spaces of operators on them. The author was supported in part by the NSF grant DMS-9970369.