Schatten P-classes Research Articles

Clarkson-McCarthy inequalities with unitary and isometry orbits

A refinement of a trace inequality of McCarthy establishing the uniform convexity of the Schatten p-classes for p>2 is proved: if A,B are two n-by-n matrices, then there exists some pair of n-by-n unitary matrices U,V such thatU|A+B2|pU⁎+V|A−B2|pV⁎≤|A|p+|B|p2. A similar statement holds for compact Hilbert space operators. Another improvement of McCarthy's inequality is given via the new operator parallelogramm law,|A+B|2⊕|A−B|2=U0(|A|2+|B|2)U0⁎+V0(|A|2+|B|2)V0⁎ for some pair of 2n-by-n isometry matrices U0,V0.

Reduced spectral synthesis and compact operator synthesis

We introduce and study the notion of reduced spectral synthesis, which unifies the concepts of spectral synthesis and uniqueness in locally compact groups. We exhibit a number of examples and prove that every non-discrete locally compact group with an open abelian subgroup has a subset that fails reduced spectral synthesis. We introduce compact operator synthesis as an operator algebraic counterpart of this notion and link it to other exceptional sets in operator algebra theory, studied previously. We show that a closed subset E of a second countable locally compact group G satisfies reduced local spectral synthesis if and only if the subset E⁎={(s,t):ts−1∈E} of G×G satisfies compact operator synthesis. We apply our results to questions about the equivalence of linear operator equations with normal commuting coefficients on Schatten p-classes.

Time–Frequency Localization Operators and a Berezin Transform

Time–frequency localization operators are a quantization procedure that maps symbols on \({\mathbb{R}^{2d}}\) to operators and depends on two window functions. We study the range of this quantization and its dependence on the window functions. If the If the short-time Fourier transform of the windows does not have any zero, then the range is dense in the Schatten p-classes. The main tool is new version of the Berezin transform associated to operators on \({L^{2}(\mathbb{R}^{d})}\). Although some results are analogous to results about Toeplitz operators on spaces of holomorphic functions, the absence of a complex structure requires the development of new methods that are based on time-frequency analysis.

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|) (see Fig. 1). We obtain an explicit characterization for the membership in Schatten p-classes, in terms of γ.

Hankel operators on large weighted Bergman spaces

We completely describe the boundedness, compactness and membership in Schatten p-classes of the (big) Hankel operator with conjugate analytic symbols on large weighted Bergman spaces.

Eigenvalue decay rates for positive integral operators

We present decay rates for the eigenvalues of positive integral operators with smooth kernels on special metric spaces endowed with a strictly positive measure. The smoothness is defined by either differentiability conditions or inequalities of Lipschitz type. We use the decay rates to place the operators in some Schatten p-classes.

Hankel Operators on Standard Bergman Spaces

We study Hankel operators on the standard Bergman spaces \({A^2_{\alpha},\,\alpha > -1}\). A description of the boundedness and compactness of the (big) Hankel operator H f with general symbols \({f \in L^2(\mathbb{D}, dA_{\alpha})}\) is obtained. Also, we provide a new proof of a result of Arazy–Fisher–Peetre on the membership in Schatten p-classes of Hankel operators with conjugate analytic symbols.

Toeplitz Operators on the Fock Space

We study Toeplitz operators on the Fock space with positive measures as symbols. Main results include characterizations of Fock–Carleson measures, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten p-classes.

On Axiomatizability of Non-Commutative Lp-Spaces

AbstractIt is shown that Schatten p-classes of operators between Hilbert spaces of different (infinite) dimensions have ultrapowers which are (completely) isometric to non-commutative Lp-spaces. On the other hand, these Schatten classes are not themselves isomorphic to non-commutative Lp spaces. As a consequence, the class of non-commutative Lp-spaces is not axiomatizable in the first-order language developed by Henson and Iovino for normed space structures, neither in the signature of Banach spaces, nor in that of operator spaces. Other examples of the same phenomenon are presented that belong to the class of corners of non-commutative Lp-spaces. For p = 1 this last class, which is the same as the class of preduals of ternary rings of operators, is itself axiomatizable in the signature of operator spaces.

Double Operator Integrals

This paper is concerned with perturbation formulae of the form∥f(a)−f(b)∥Lp(M,τ)⩽K∥a−b∥ Lp(M,τ) with K>0 being a constant depending on p and f only, where f is a real-valued Lipschitz function and a,b are self-adjoint τ-measurable operators affiliated with a semifinite von Neumann algebra (M,τ), such that the difference a−b belongs to Lp(M,τ), 1<p<∞. In order to treat the situation where the von Neumann algebra M is not necessarily hyperfinite, we first develop an integration theory with respect to finitely additive spectral measures in a Banach space. Applied to product measures this integration theory may be considered as an abstract version of the double operator integrals due to Birman and Solomyak. To describe the class of integrable functions we employ our recent study of multiplier theory in UMD-spaces. Our perturbation formulae extend those of Davies and Birman–Solomyak for the case when M is a hyperfinite I∞-factor (i.e., for the Schatten p-classes). We also discuss analogous perturbation results in the setting of symmetric operator spaces associated with (M,τ).

Range-kernel orthogonality of the elementary operator X→∑ i=1 nAiXBi− X

Let H be a separable infinite dimensional complex Hilbert space and let B( H) denote the algebra of operators on H into itself. Let A =(A 1,A 2,…,A n) and B =(B 1,B 2,…,B n) be n-tuples in B( H). Define the elementary operators ▵ AB and ▵ * AB :B(H)→B(H) by ▵ AB (X)=∑ i=1 nA iXB i−X and ▵ * AB (X)=∑ i=1 nA i *XB i *−X . This note considers the range-kernel orthogonality of the restrictions of ▵ AB and ▵ * AB to Schatten p-classes C p . It is proved that: (a) if 1< p<∞ , S∈ C p and ∑ i=1 n A * i A i , ∑ i=1 n A i A * i , ∑ i=1 n B * i B i and ∑ i=1 n B i B * i are all ⩽1, then min{∥▵ AB (X)+S∥ p,∥▵ * AB (X)+S∥ p}⩾∥S∥ p for all X∈ C p if and only if ▵ AB (S)=0=▵ * AB (S) ; (b) if p=2 and S∈ C 2 , then ∥▵ AB (X)+S∥ 2 2=∥▵ AB (X)∥ 2 2+∥S∥ 2 2 and ∥▵ * AB (X)+S∥ 2 2=∥▵ * AB (X)∥ 2 2+∥S∥ 2 2 if and only if ▵ AB (S)=0=▵ * AB (S) ; and (c) if A and B are the n-tuples of (a) such that ▵ BB (S)=0=▵ * BB (S) for some injective S∈ C 1 , then the inequality of (a) holds (with p=1 and) for all X∈ C 1 if and only if ▵ AB (S)=0=▵ * AB (S) .

Reflexivity of the group of surjective isometries on some Banach spaces

In this paper we study the problem of algebraic reflexivity of the isometry group of some important Banach spaces. Because of the previous work in similar topics, our main interest lies in the von Neumann – Schatten p-classes of compact operators. The ideas developed there can be used in ℓp-spaces, Banach spaces of continuous functions and spin factors as well. Moreover, we attempt to attract the attention to this problem from general Banach spaces geometry view-point. This study, we believe, would provide nice geometrical results.

Boundedness, compactness, and Schatten p-classes of Hankel operators between weighted Dirichlet spaces

Boundedness, compactness, and Schatten p-classes of Hankel operators between weighted Dirichlet spaces

Separable representations for automorphism groups of infinite symmetric spaces

In this paper we consider the separable unitary representations for the automorphism groups of the classical infinite rank (Finsler) symmetric spaces defined by Schatten p-classes (often referred to as restricted groups). Following earlier work of Ol'shanskii and Voiculescu, it is shown that the spherical representations are always type I, the form of the irreducible spherical functions is determined, and their analyticity established. Using an intuitive geometric argument, it is shown that the real spherical functions extend to the Hilbert-Schmidt limit and never beyond. This yields a complete determination of the separable representations for groups corresponding to p-classes with p > 2.