On Hille-Tamarkin operators and Schatten classes

We determine the smallest Schatten class containing all integral operators with kernels inL _p(L_{p', q})^symm, where 2 <p∞ and 1≦q≦∞. In particular, we give a negative answer to a problem posed by Arazy, Fisher, Janson and Peetre in [1].     

Differential equations in Schatten classes of operators

We study a system of differential equations in Schatten classes of operators, ${\mathcal{C}_p(\mathcal{H})\,(1 \leq p < \infty}$ ), with ${\mathcal{H}}$ a separable complex Hilbert space. The systems considered are infinite dimensional generalizations of mathematical models of unsupervised learning. In this new setting, we address the usual questions of existence and uniqueness of solutions. Under some restrictions on the spectral properties of the initial conditions, we explicitly solve the system. We also discuss the long-term behavior of solutions.     

Schatten classes of integration operators on Dirichlet spaces

We address the problem of determining membership in Schatten-Von Neumann ideals S _p of integration operators (T _g f)(z) = ∫ ₀ ^z = ∫ ₀ ^z f(ξ)g′(ξ)dξ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators [10, p. 347].     

Composition operators with univalent symbol in Schatten classes

We study composition operators, induced by a sub-domain of the unit disc whose boundary intersects the unit circle at 1 and which has, in a neighborhood of 1, a polar equation 1−r=γ(|θ|)$1-r=\gamma (|\theta |)$ (see Fig. 1). We obtain an explicit characterization for the membership in Schatten p-classes, in terms of γ.     

Schatten classes for Toeplitz operators with Hilbert space windows on modulation spaces

We prove that if ω, ω₁, ω₂, v₁, v₂ are appropriate, , j=1,2, and ωa∈Lp, then the Toeplitz operator Tph₁,h₂(a) from to belongs to the Schatten-von Neumann class of order p. From this property we prove convolution properties between weighted Lebesgue spaces and Schatten-von Neumann classes of symbols in pseudo-differential calculus.     

Multipliers of Schatten classes

Let C_p be the class of all compact operators A on the Hilbert space l² for which $\text{∑¦λ}{}_{\mathrm{i}}\text{¦}{}^{\mathrm{p}}\text{< ∞}$, where {λ_i} is the eigen values of $\text{(A}{}^1\text{A)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. The object of this paper is to prove some results concerning the Schur multipliers of C_p.     

Hankel operators in Schatten ideals

We study membership to Schatten ideals S _E , associated with a monotone Riesz–Fischer space E, for the Hankel operators H _f defined on the Hardy space H ²(∂D). The conditions are expressed in terms of regularity of its symbol: we prove that H _f ∈S _E if and only if f∈B _E , the Besov space associated with a monotone Riesz–Fischer space E(dλ) over the measure space (D,dλ) and the main tool is the interpolation of operators. Received: December 17, 1999; in final form: September 25, 2000?Published online: July 13, 2001     

Concentration of mass on the Schatten classes

Let 1?p?∞ and be the unit ball of the Schatten trace class of matrices on Cn or on Rn, normalized to have Lebesgue measure equal to one. We prove that

     

On some classes of unbounded operators

Hyponormality, normality and subnormality for unbounded operators on Hilbert space are investigated and quasi- similarity of such operators is discussed.     

Schatten classes on compact manifolds: Kernel conditions

In this paper we give criteria on integral kernels ensuring that integral operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given.     

Schatten class Toeplitz operators on generalized Fock spaces

In this paper we characterize the Schatten p   class membership of Toeplitz operators with positive measure symbols acting on generalized Fock spaces for the full range 0<p<∞$0<p<\infty$.     

Schatten class hankel operators on the Bergman space

In this paper we characterize Hankel operatorsH _f andH _f on the Bergman spaces of bounded symmetric domains which are in the Schatten p-class for 2p< and f inL ² using a Jordan algebra characterization of bounded symmetric domains and properties of the Bergman metric.     

Paraproducts and Hankel operators of Schatten class via p-John-Nirenberg Theorem

We give an interpolation-free proof of the known fact that a dyadic paraproduct is of Schatten-von Neumann class S_p, if and only if its symbol is in the dyadic Besov space B_p^d. Our main tools are a product formula for paraproducts and a “p-John-Nirenberg-Theorem” due to Rochberg and Semmes.We use the same technique to prove a corresponding result for dyadic paraproducts with operator symbols.Using an averaging technique by Petermichl, we retrieve Peller's characterizations of scalar and vector Hankel operators of Schatten-von Neumann class S_p for 1<p<∞. We then employ vector techniques to characterise little Hankel operators of Schatten-von Neumann class, answering a question by Bonami and Peloso.Furthermore, using a bilinear version of our product formula, we obtain characterizations for boundedness, compactness and Schatten class membership of products of dyadic paraproducts.