Schatten Spaces Research Articles

2-positive contractive projections on noncommutative Lp-spaces

We prove the first theorem on projections on general noncommutative Lp-spaces associated with non-type I von Neumann algebras where 1⩽p<∞. This is the first progress on this topic since the seminal work of Arazy and Friedman (1992) [6] where the problem of the description of contractively complemented subspaces of noncommutative Lp-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative Lp-space is completely order isometrically isomorphic to some noncommutative Lp-space. This result is sharp and is even new for Schatten spaces Sp. Our approach relies on non-tracial Haagerup's noncommutative Lp-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.

Improved spectral cluster bounds for orthonormal systems

Abstract We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ {p=\infty} , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , ( λ + 1 ) 2 ) {[\lambda^{2},(\lambda+1)^{2})} to [ λ 2 , ( λ + ϵ ⁢ ( λ ) ) 2 ) {[\lambda^{2},(\lambda+\epsilon(\lambda))^{2})} , where ϵ ⁢ ( λ ) {\epsilon(\lambda)} is a function of λ that goes to 0 as λ goes to ∞ {\infty} . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p L^{p} -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527].

Contractively decomposable projections on noncommutative Lp-spaces

We describe and characterize the contractively decomposable projections on noncommutative Lp-spaces. Our result relies on a new lifting result for decomposable maps of independent interest and on some tools from ergodic theory. Our theorem is new even for finite-dimensional Schatten spaces. Our description allows us to connect this topic with W⁎-ternary rings of operators and a slight generalization of our result for more general projections makes JBW⁎-triples appear in this context. We also prove that all rectangular Lp-spaces associated with W⁎-ternary rings of operators arise as contractively decomposable complemented subspaces of noncommutative Lp-spaces. Finally, we introduce a notion of Lp-space associated to each σ-finite JBW⁎-triple and we explain the link with the context of this paper.

Dilations of Markovian semigroups of measurable Schur multipliers

AbstractUsing probabilistic tools, we prove that any weak* continuous semigroup $(T_t)_{t \geqslant 0}$ of self-adjoint unital completely positive measurable Schur multipliers acting on the space $\mathrm {B}({\mathrm {L}}^2(X))$ of bounded operators on the Hilbert space ${\mathrm {L}}^2(X)$ , where X is a suitable measure space, can be dilated by a weak* continuous group of Markov $*$ -automorphisms on a bigger von Neumann algebra. We also construct a Markov dilation of these semigroups. Our results imply the boundedness of the McIntosh’s ${\mathrm {H}}^\infty $ functional calculus of the generators of these semigroups on the associated Schatten spaces and some interpolation results connected to ${\mathrm {BMO}}$ -spaces. We also give an answer to a question of Steen, Todorov, and Turowska on completely positive continuous Schur multipliers.

On Arens regularity of projective tensor product of Schatten spaces

We discuss Arens regularity of the projective tensor product of Banach algebras consisting of Schatten class operators. More precisely, exploiting Ülger’s biregularity criterion, we prove that for any Hilbert space ℋ, the Banach algebras Sp(ℋ)⊗γ Sq (ℋ) and B(S2(ℋ))⊗γ S2(ℋ) are not Arens regular for every pair 1≤p,q≤2; whereas, when ℋ is separable and S2(ℋ) is equipped with the Schur product, we show that S2(ℋ) is completely continuous and that S2(ℋ)⊗γ S2(ℋ) is Arens regular.

Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces

We generalize the Lp spectral cluster bounds of Sogge for the Laplace–Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.

Restriction theorems for orthonormal functions, Strichartz inequalities, and uniform Sobolev estimates

We generalize the theorems of Stein-Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schr\odinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb, and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-posedness of the Hartree equation in Schatten spaces, to Lieb-Thirring bounds for eigenvalues of Schr\odinger operators with complex potentials, and to Schatten properties of the scattering matrix.

Eigenvalue bounds for Dirac and fractional Schrödinger operators with complex potentials

We prove Lieb–Thirring type bounds for fractional Schrödinger operators and Dirac operators with complex-valued potentials. The main new ingredient is a resolvent bound in Schatten spaces for the unperturbed operator, in the spirit of Frank and Sabin [11].

The Stein-Tomas inequality in trace ideals

The goal of this review is to explain some recent results [5] regarding generalizations of the Stein-Tomas (and Strichartz) inequalities to the context of trace ideals (Schatten spaces).

Local smoothing for the quantum Liouville equation

We analyze in this work the regularity properties of the density operator solution to the quantum Liouville equation. As was recently done for the Strichartz inequalities, we extend to systems of orthonormal functions the local smoothing estimates satisfied by the solutions to the Schrödinger equation. We show in particular that the local density associated to the solution to the free, linear, quantum Liouville equation admits locally fractional derivatives of given order provided the data belong to some Schatten spaces.

Strichartz inequality for orthonormal functions

We prove a Strichartz inequality for a system of orthonormal functions, with an optimal behavior of the constant in the limit of a large number of functions. The estimate generalizes the usual Strichartz inequality, in the same fashion as the Lieb-Thirring inequality generalizes the Sobolev inequality. As an application, we consider the Schrödinger equation in a time-dependent potential and we show the existence of the wave operator in Schatten spaces.

All Schatten spaces endowed with the Schur product are Q-algebras

We prove that the Banach algebra formed by the space of compact operators on a Hilbert space endowed with the Schur product is a quotient of a uniform algebra (also known as a Q-algebra). Together with a similar result of Pérez-García for the trace class, this completes the answer to a long-standing question of Varopoulos.

A new characterization of Bergman–Schatten spaces and a duality result

Let B 0 ( D , ℓ 2 ) denote the space of all upper triangular matrices A such that lim r → 1 − ( 1 − r 2 ) ‖ ( A ∗ C ( r ) ) ′ ‖ B ( ℓ 2 ) = 0 . We also denote by B 0 , c ( D , ℓ 2 ) the closed Banach subspace of B 0 ( D , ℓ 2 ) consisting of all upper triangular matrices whose diagonals are compact operators. In this paper we give a duality result between B 0 , c ( D , ℓ 2 ) and the Bergman–Schatten spaces L a 1 ( D , ℓ 2 ) . We also give a characterization of the more general Bergman–Schatten spaces L a p ( D , ℓ 2 ) , 1 ⩽ p < ∞ , in terms of Taylor coefficients, which is similar to that of M. Mateljevic and M. Pavlovic [M. Mateljevic, M. Pavlovic, L p -behaviour of the integral means of analytic functions, Studia Math. 77 (1984) 219–237] for classical Bergman spaces.

Completely 1-complemented subspaces of Schatten spaces

We consider the Schatten spaces S p S^p in the framework of operator space theory and for any 1 ≤ p ≠ 2 &gt; ∞ 1\leq p\not =2&gt;\infty , we characterize the completely 1 1 -complemented subspaces of S p S^p . They turn out to be the direct sums of spaces of the form S p ( H , K ) S^p(H,K) , where H , K H,K are Hilbert spaces. This result is related to some previous work of Arazy and Friedman giving a description of all 1 1 -complemented subspaces of S p S^p in terms of the Cartan factors of types 1–4. We use operator space structures on these Cartan factors regarded as subspaces of appropriate noncommutative L p L^p -spaces. Also we show that for any n ≥ 2 n\geq 2 , there is a triple isomorphism on some Cartan factor of type 4 and of dimension 2 n 2n which is not completely isometric, and we investigate L p L^p -versions of such isomorphisms.

$L^p$-matricially normed spaces and operator space valued Schatten spaces

Let 1 < p < ∞ and let F be an operator space. Let S p k [F] be Pisier's operator space valued Schatten space, for any integer k ≥ 1. Then F equipped with the matrix norms given by the S p k [F]'s is an L p -matricially normed space. We show that if p ≠ 1, not all L p -matricially normed spaces are of this form. Then we give a characterization of those L p -matricially normed spaces which are of this form.

Riesz transforms on generalized Heisenberg groups and Riesz transforms associated to the CCR heat flow

Let 1 < q < ∞. We prove that the Riesz transforms Rk = XkL − 1 2 on a generalized Heisenberg group G satisfy PK k=1 |Rk(f)|2 1 2 Lq (G) ≤ C(q, J) kfkLq(G) where K, J are respectively the dimensions of the first and second layer of the Lie algebra of G. We prove similar inequalities on Schatten spaces Sq(H), with dimension free constants, for Riesz transforms associated to commuting inner ∗-derivations Dk and a suitable substitute of the square function. An example is given by the derivations associated to n commuting pairs of operators (Pj , Qj ) on a Hilbert space H satisfying the canonical commutation relations [Pj , Qj ] = iIH.

Riesz transforms on generalized Heisenberg groups and Riesz transforms

Let $1 < q < \infty$. We prove that the Riesz transforms $R_{k}=X_{k} L^{-\frac{1}{2}}$ on a generalized Heisenberg group $G$ satisfy $\left\|\left(\sum_{k=1}^{K}\left| R_{k}(f)\right| ^{2}\right)^{\frac{1}{2}}\right\| _{L^{q}(G)}\leq C(q,J)\left\| f\right\| _{L^{q}(G)}$ where $K$, $J$ are respectively the dimensions of the first and second layer of the Lie algebra of $G$. We prove similar inequalities on Schatten spaces $S^{q}(H)$, with dimension free constants, for Riesz transforms associated to commuting inner $*$-derivations $D_{k}$ and a suitable substitute of the square function. An example is given by the derivations associated to $n$ commuting pairs of operators $(P_{j},Q_{j})$ on a Hilbert space $H$ satisfying the canonical commutation relations [P$_{j},Q_{j}]=iI_{H}$.