Completely p-summing maps on the operator Hilbert space OH

We prove the completely p-summing ideals of OH are all equal as sets for 1?p<2. A phase transition then occurs at p=2 as we also show for p?2, the completely p-summing ideals of OH turn out as sets to be Schatten ideal classes with the limiting case being the Schatten 4-class ideal S₄ when p→∞.     

Interpolation of operator ideals with an application to eigenvalue distribution problems

We derive results on the interpolation of complete quasinormed operator ideals, mainly for the absolutelyp-summing and thes-number idealsS _p ^s defined by Pietsch. By estimating theK-Functional of Peetre, we get that the interpolation ideal (S _p1 ^s ,S _p2 ^s ),_,p is contained inS _p ^s and is even equal to it in the case of the approximation numbers. A similar fact is proved for absolutely (p, q)-summing operators, interpolating the first index. We show further that the absolutelyp-summing operators onc ₀ are contained in the complex interpolation space ( _p1 (c _o), _p2 (c _o))_[].The previous results are then applied to prove summability properties for the eigenvalues of operators in Banach spaces, which are products ofS _p1 ^s -type and absolutelyp _j -summing operators. Roughly speaking, the summability order is the harmonic sum of thep _i - andp _j -indices, wherep _j 2. In the case of Hilbert spaces, this reduces to the well-known Weyl-inequality. The method uses an abstract interpolation estimate for ideal quasinorms which may be useful also for other operator ideals.     

Domination problems for strictly singular operators and other related classes

We survey recent results on domination properties of strictly singular operators and related operator ideals, as well as Banach–Saks operators, Narrow operators and p-summing operators.     

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

Factorization of Hardy spaces and Hankel operators on convex domains in ℂ n

In this article we study the (small) Hankel operator h_b on the Hardy and Bergman spaces on a smoothly bounded convex domain of finite type in ℂⁿ. We completely characterize the Hankel operators h_b that are bounded, compact, and belong to the Schatten ideal S_p, for 0 < p < ∞. In particular, if h_b denotes the Hankel operator on the Hardy space H² (Ω), we prove that h_b is bounded if and only if b ∈ BMOA, compact if and only if b ∈ VMOA, and in the Schatten class if and only if b ∈e B_p, 0 < p < ∞. This last result extends the analog theorem in the case of the unit disc of Peller [19] and Semmes [21]. In order to characterize the bounded Hankel operators, we prove a factorization theorem for functions in H¹ (Ω), a result that is of independent interest.     

A Factorization Property of R‐Bounded Sets of Operators on Lp‐Spaces

Let T be a bounded operator on L^p‐space, with 1 ≤ p < ∞. A theorem of W. B. Johnson and L. Jones asserts that after an appropriate change of density, T actually extends to a bounded operator on L². We show that if 𝒯 ⊂ B (L^p) is an R‐bounded set of operators, then the latter result holds for any T ∈ 𝒯 with a common change of density. Then we give applications including results on R‐sectorial operators.     

A Note on the Spectrum of Invertible p-hyponormal Operators

A bounded linear operator T is clalled p-hyponormal if (T*T)^p ≥ (TT)^p, 0 < p < 1. It is known that for semi-hyponormal operators (p = 1/2), the spectrum of the operator is equal to the union of the spectra of the general polar symbols of the operator. In this paper we prove a somewhat weaker result for invertible p-hyponormal operators for 0 < p < 1/2.     

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

Multiple summing operators on<Emphasis Type="Italic">C(K)</Emphasis> spaces

In this paper, we characterize, for 1≤p<∞, the multiple (p, 1)-summing multilinear operators on the product ofC(K) spaces in terms of their representing polymeasures. As consequences, we obtain a new characterization of (p, 1)-summing linear operators onC(K) in terms of their representing measures and a new multilinear characterization ofL _∞ spaces. We also solve a problem stated by M.S. Ramanujan and E. Schock, improve a result of H. P. Rosenthal and S. J. Szarek, and give new results about polymeasures. Both authors were partially supported by DGICYT grant BMF2001-1284.     

Ideals of a C *-algebra generated by an operator algebra

In this paper, we consider ideals of a C ^*-algebra C^*(B){C^*(\mathcal{B})} generated by an operator algebra B{\mathcal{B}} . A closed ideal J í C^*(B){J\subseteq C^*(\mathcal{B})} is called a K-boundary ideal if the restriction of the quotient map on B{\mathcal{B}} has a completely bounded inverse with cb-norm equal to K ⁻¹. For K = 1 one gets the notion of boundary ideals introduced by Arveson. We study properties of the K-boundary ideals and characterize them in the case when operator algebra λ-norms itself. Several reformulations of the Kadison similarity problem are given. In particular, the affirmative answer to this problem is equivalent to the statement that every bounded homomorphism from C^*(B){C^*(\mathcal{B})} onto B{\mathcal{B}} which is a projection on B{\mathcal{B}} is completely bounded. Moreover, we prove that Kadison’s similarity problem is decided on one particular C ^*-algebra which is a completion of the *-double of M₂(\mathbbC){M_2(\mathbb{C})} .     

The Finite Hilbert Transform in ℒ2

It is well known that the finite HILBERT transform T is a NOETHER (FREDHOLM) operator when considered as a map from ?^p into itself if 1 < p < 2 or 2 < p < ∞. When p = 2, the map T is not a NOETHER operator. We present two theorems which characterize the range of T in ?² and, as immediate consequences, give simple expressions for its inverse.     

On the existence of principal values for the cauchy integral on weighted lebesgue spaces for non-doubling measures

Let T be a Calderón-Zygmund operator in a “non-homogeneous” space ( , d, μ), where, in particular, the measure μ may be non-doubling. Much of the classical theory of singular integrals has been recently extended to this context by F. Nazarov, S. Treil, and A. Volberg and, independently by X. Tolsa. In the present work we study some weighted inequalities for T_*, which is the supremum of the truncated operators associated with T. Specifically, for1<p<∞, we obtain sufficient conditions for the weight in one side, which guarantee that another weight exists in the other side, so that the corresponding L^p weighted inequality holds for T_*. The main tool to deal with this problem is the theory of vector-valued inequalities for T_* and some related operators. We discuss it first by showing how these operators are connected to the general theory of vector-valued Calderón-Zygmund operators in non-homogeneous spaces, developed in our previous paper [6]. For the Cauchy integral operator C, which is the main example, we apply the two-weight inequalities for C_* to characterize the existence of principal values for functions in weighted L^p.     

Maurey’s factorization theory for operator spaces

We prove an operator space version of Maurey’s theorem, which claims that every absolutely (p, 1)-summing map on C(K) is automatically absolutely q-summing for q > p. Our results imply in particular that every completely bounded map from B(H) with values in Pisier’s operator space OH is completely p-summing for p > 2. This fails for p = 2. As applications, we obtain eigenvalue estimates for translation invariant maps defined on the von Neumann algebra V N(G) associated with a discrete group G. We also develop a notion of cotype which is compatible with factorization results on noncommutative L _p spaces.     

Cohen and multiple Cohen strongly summing multilinear operators

Considering the successful theory of multiple summing multilinear operators as a prototype, we introduce the classes of multiple Cohen strongly p-summing multilinear operators and polynomials. The adequacy of these classes under the viewpoint of the theory of multilinear and polynomial ideals and holomorphy types is discussed in detail.     

Computingp-summing norms with few vectors

It is shown that thep-summing norm of any operator withn-dimensional domain can be well-aproximated using only “few” vectors in the definition of thep-summing norm. Except for constants independent ofn and logn factors, “few” meansn if 1<p<2 andn ^p/2 if 2<p<∞. Supported in part by NSF #DMS90-03550 and the U.S.-Israel Binational Science Foundation. Supported in part by the U.S.-Israel Binational Science Foundation.     

Isolation and Component Structure in Spaces of Composition Operators

We establish a condition that guarantees isolation in the space of composition operators acting between H^p(B_N) and H^q(B_N), for 0 < p ≤ ∞, 0 < q < ∞, and N ≥ 1. This result will allow us, in certain cases where 0 < q < p ≤ ∞, completely to characterize the component structure of this space of operators.     

On Ideal Operators

Let E, F be Archimedean Riesz spaces. We consider operators that map ideals of E to ideals of F and operators T for which, T ^–1 (I) is an ideal in E, for each ideal I in F. We study the properties of such operators and investigate their relation to disjointness preserving operators.     

Antisymmetric tensor products of absolutely p-summing operators

We consider antisymmetric tensor products of absolutely p-summing operators. In connection with this second moments of determinants of random matrices appear. These second moments are closely related to approximation properties of the absolutely 2-summing operators and can be used to characterize some classes of infinite-dimensional Banach spaces. Finite-dimensional results are also obtained by this approach.     

Compact operators which factor through subspaces of lp

Let 1 ≤ p ≤ ∞. A subset K of a Banach space X is said to be relatively p ‐compact if there is an 〈x_n 〉 ∈ l^s _p (X) such that for every k ∈ K there is an 〈α_n 〉 ∈ l_{p ′} such that k = σ^∞_n=1 α_n x_n . A linear operator T: X → Y is said to be p ‐compact if T (Ball (X)) is relatively p ‐compact in Y. The set of all p ‐compact operators K_p (X, Y) from X to Y is a Banach space with a suitable factorization norm κ_p and (K_p , κ_p ) is a Banach operator ideal. In this paper we investigate the dual operator ideal (K^d _p , κ^d _p ). It is shown that κ^d _p (T) = π_p (T) for all T ∈ B (X, Y) if either X or Y is finite‐dimensional. As a consequence it is proved that the adjoint ideal of K^d _p is I_{p ′}, the ideal of p ′‐integral operators. Further, a composition/decomposition theorem K^d _p = Π_p K is proved which also yields that (Π^min _p )^inj = K^d _p . Finally, we discuss the density of finite rank operators in K^d _p and give some examples for different values of p in this respect. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Block Matrix Operators and Weak Hyponormalities

We introduce a new model of a block matrix operator M(α, β) induced by two sequences α and β and characterize its p-hyponormality. The model may be viewed as arising from the composition operator C_T on l²₊ : = L²(\mathbb N₀)l^{2}_{+} := L^2(\mathbb {N}_0) induced by a measurable transformation T on the set of nonnegative integers \mathbb N₀\mathbb {N}_0 with point mass measure. Composition operator techniques may then be used to treat the p-hyponormality of M(α, β). Finally, we apply our results to obtain examples of these operators showing the p-hyponormal classes are distinct.