Approximation numbers of weighted composition operators

A general method for estimating the approximation numbers of composition operators on the Hardy space H 2, using finite-dimensional model subspaces, is studied and applied in the case when the symbol of the operator maps the unit disc to a domain whose boundary meets the unit circle at just one point. The exact rate of decay of the approximation numbers is identified when this map is sufficiently smooth at the point of tangency; it follows that a composition operator with any prescribed slow decay of its approximation numbers can be explicitly constructed. Similarly, an asymptotic expression for the approximation numbers is found when the mapping has a sharp cusp at the distinguished boundary point. Precise asymptotic estimates in the intermediate cases, including that of maps with a corner at the distinguished boundary point, are also established.

Complex symmetry of invertible composition operators

The topological structure of the set of (weighted) composition operators has been studied on various function spaces on the unit disc such as Hardy spaces, the space of bounded holomorphic functions, weighted Banach spaces of holomorphic functions with sup-norm, Hilbert Bergman spaces. In this paper we consider this problem for all Bergman spaces $$A_{\alpha }^p$$ with $$p \in (0, \infty )$$ and $$ \alpha \in (-1, \infty )$$ . In this setting we establish a criterion for two composition operators to be linearly connected in the space of composition operators; furthermore, for the space of weighted composition operators, we prove that the set of compact weighted composition operators is path connected, but it is not a component.

Relative entropy and curved spacetimes

We present a unified approach to the problem of characterizing the Fredholm multiplication and composition operators on the Hardy spaceH 2.

We consider quantum information tasks in an operator algebraic setting, where we consider normal states on von Neumann algebras. In particular, we consider subfactors N ⊂ M \mathfrak {N} \subset \mathfrak {M} , that is, unital inclusions of von Neumann algebras with trivial center. One can ask the following question: given a normal state ω \omega on M \mathfrak {M} , how much can one learn by only doing measurements from N \mathfrak {N} ? We argue how the Jones index [ M : N ] [\mathfrak {M}:\mathfrak {N}] can be used to give a quantitative answer to this, showing how the rich theory of subfactors can be used in a quantum information context. As an example we discuss how the Jones index can be used in the context of wiretap channels. Subfactors also occur naturally in physics. Here we discuss two examples: rational conformal field theories and Kitaev’s toric code on the plane, a prototypical example of a topologically ordered model. There we can directly relate aspects of the general setting to physical properties such as the quantum dimension of the excitations. In the example of the toric code we also show how we can calculate the index via an approximation with finite dimensional systems. This explicit construction sheds more light on the connection between topological order and the Jones index.

There is axiomatically described the class of spaces Υ (resp. Χ) of functions, analytic in the unit disk, for which the invariant subspaces of the shift operator f (z) → z f (z) (resp. the inverse shift f(z)→z−1(f(z)−f (0))) are constructed just like the Hardy space H2. It is proved that as Χ one can take, for example, the space H1, the disk-algebra CA, the space UA of all uniformly convergent power series; and as Υ the space of integrals of Cauchy type L1/H 1− , the space VMOA. There is also obtained an analog for the space UA of W. Rudin's theorem on z-invariant subspaces of the space CA.

Suppose $\\varphi$ is an analytic self-map of open unit disk $\\mathbf{D}$ and $\\psi$ is an analytic function on $\\mathbf{D}$. Then a weighted composition operator induced by $\\varphi$ with weight $\\psi$ is given by $(W_{\\psi, \\varphi}f)(z)= \\psi(z)f(\\varphi(z))$, for $z \\in \\mathbf{D}$ and $f$ analytic on $\\mathbf{D}$. Necessary and sufficient conditions are given for the boundedness and compactness of the weighted composition operators $W_{\\psi, \\varphi}$. In terms of fixed points in the closed unit disk $\\overline{\\mathbf{D}}$, conditions under which $W_{\\psi, \\varphi}$ is compact are given. Necessary conditions for the compactness of $C_{\\varphi}$ are given in terms of the angular derivative $\\varphi^{\\prime}(\\zeta)$ where $\\zeta$ is on the boundary of the unit disk. Moreover, we present sufficient conditions for the membership of composition operators in the Schatten $p$-class $S_{p}(H^s(\\beta_1), H^q(\\beta_2))$, where the inducing map has supremum norm strictly smaller than~$1$.

Toeplitz and asymptotic Toeplitz operators on [formula omitted

In this paper, we study 2-complex symmetric composition operators with the conjugation J, defined by Jf(z)=(f(z¯))¯, on the Hardy space H2. More precisely, we obtain the necessary and sufficient condition for the composition operator Cϕ to be 2-complex symmetric with J when ϕ is an automorphism of D. We also characterize 2-complex symmetric with J when ϕ is a linear fractional self-map of D.

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

We introduce $L^2$-Betti numbers, as well as a general homology and cohomology theory for the standard invariants of subfactors, through the associated quasi-regular symmetric enveloping inclusion of II_1 factors. We actually develop a (co)homology theory for arbitrary quasi-regular inclusions of von Neumann algebras. For crossed products by countable groups $Γ$, we recover the ordinary (co)homology of $Γ$. For Cartan subalgebras, we recover Gaboriau's $L^2$-Betti numbers for the associated equivalence relation. In this common framework, we prove that the $L^2$-Betti numbers vanish for amenable inclusions and we give cohomological characterizations of property (T), the Haagerup property and amenability. We compute the $L^2$-Betti numbers for the standard invariants of the Temperley-Lieb-Jones subfactors and of the Fuss-Catalan subfactors, as well as for free products and tensor products.

It is proved that given a separable von Neumann algebra $A$ which contains a Cartan subalgebra $D$, there always exists, for any intermediate von Neumann subalgebra $B$ with $D\subseteq B$, a faithful normal conditional expectation from $A$ onto $B$.

The composition operator induced by a holomorphic self-map of the unit disc is compact on L 1 {L^1} of the unit circle if and only if it is compact on the Hardy space H 2 {H^2} of the disc. This answers a question posed by Donald Sarason: it proves that Sarason’s integral condition characterizing compactness on L 1 {L^1} is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on H 2 {H^2} .

Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions