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In this paper it is shown that Toeplitz operators on Bergman space form a dense subset of the space of all bounded linear operators, in the strong operator topology, and that their norm closure contains all compact operators. Further, the$C$^{*}-algebra generated by them does not contain all bounded operators, since all Toeplitz operators belong to the essential commutant of certain shift. The result holds in Bergman spaces$A$^{2}(Ω) for a wide class of plane domains Ω⊂$C$, and in Fock spaces$A$^{2}($C$^{N}),$N$≧1.

We prove that intermediate Banach spaces $$\mathcal{A}$$ and $$\mathcal{B}$$ with respect to arbitrary Hilbert couples $$\bar {H}$$ and $$\bar {K}$$ are exact interpolation if and onlyif they are exact$K$-monotonic, i.e. the condition $$f^0 \in \mathcal{A}$$ and the inequality $$K(t,g^0 ;\bar {K}) \leqslant K(t,f^0 ;\bar {H}),t > 0$$ , imply$g$^{0}∈$B$and ‖$g$^{0}‖$B$≤‖$f$^{0}‖_{$A$}($K$is Peetre’s$K$-functional). It is well known that this property is implied by the following: for each ϱ>1 there exists an operator $$T:\bar {H} \to \bar {K}$$ such that$Tf$^{0}=$g$^{0}, and $$K(t,Tf;\bar {K}) \leqslant \rho K(t,f;\bar {H}),f \in \mathcal{H}_0 + \mathcal{H}_1 ,t > 0$$ . Verifying the latter property, it suffices to consider the “diagonal case” where $$\bar {H} = \bar {K}$$ is finite-dimensional, in which case we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem it is shown that the statement remains valid when substituting ϱ=1. The result leads to a short proof of Donoghue’s theorem on interpolation functions, as well as Löwner’s theorem on monotone matrix functions.

This paper concerns the well-posedness theory of the motion of a physical vacuum for the compressible Euler equations with or without self-gravitation. First, a general uniqueness theorem of classical solutions is proved for the three dimensional general motion. Second, for the spherically symmetric motions, without imposing the compatibility condition of the first derivative being zero at the center of symmetry, a new local-in-time existence theory is established in a functional space involving less derivatives than those constructed for three-dimensional motions in (Coutand et al., Commun Math Phys 296:559–587, 2010; Coutand and Shkoller, Arch Ration Mech Anal 206:515–616, 2012; Jang andMasmoudi,Well-posedness of compressible Euler equations in a physical vacuum, 2008) by constructing suitable weights and cutoff functions featuring the behavior of solutions near both the center of the symmetry and the moving vacuum boundary.

For each simple Lie algebra g, we construct an algebra embedding of the quantum group Uq(g) into certain quantum torus algebra Dg via the positive representations of split real quantum group. The quivers corresponding to Dg is obtained from an amalgamation of two basic quivers, each of which is mutation equivalent to one describing the cluster structure of the moduli space of framed G-local system on a disk with 3 marked points on its boundary when G is of classical type. We derive a factorization of the universal R-matrix into quantum dilogarithms of cluster monomials, and show that conjugation by the R-matrix corresponds to a sequence of quiver mutations which produces the half-Dehn twist rotating one puncture about the other in a twice punctured disk.