Convergence structures and locally solid topologies on vector lattices of operators

For vector lattices E and F, where F is Dedekind complete and supplied with a locally solid topology, we introduce the corresponding locally solid absolute strong operator topology on the order bounded operators $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) from E into F. Using this, it follows that $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) admits a Hausdorff uo-Lebesgue topology whenever F does. For each of order convergence, unbounded order convergence, and—when applicable—convergence in the Hausdorff uo-Lebesgue topology, there are both a uniform and a strong convergence structure on $${\mathscr{L}}_{\mathrm{ob}}(E,F)$$ L ob ( E , F ) . Of the six conceivable inclusions within these three pairs, only one is generally valid. On the orthomorphisms of a Dedekind complete vector lattice, however, five are generally valid, and the sixth is valid for order bounded nets. The latter condition is redundant in the case of sequences of orthomorphisms, as a consequence of a uniform order boundedness principle for orthomorphisms that we establish. We furthermore show that, in contrast to general order bounded operators, orthomorphisms preserve not only order convergence of nets, but unbounded order convergence and—when applicable—convergence in the Hausdorff uo-Lebesgue topology as well.

Numerical solution of the infinite-dimensional LQR problem and the associated Riccati differential equations

The numerical analysis of linear quadratic regulator design problems for parabolic partial differential equations requires solving Riccati equations. In the finite time horizon case, the Riccati differential equation (RDE) arises. The coefficient matrices of the resulting RDE often have a given structure, e.g., sparse, or low-rank. The associated RDE usually is quite stiff, so that implicit schemes should be used in this situation. In this paper, we derive efficient numerical methods for solving RDEs capable of exploiting this structure, which are based on a matrix-valued implementation of the BDF and Rosenbrock methods. We show that these methods are suitable for large-scale problems by working only on approximate low-rank factors of the solutions. We also incorporate step size and order control in our numerical algorithms for solving RDEs. In addition, we show that within a Galerkin projection framework the solutions of the finite-dimensional RDEs converge in the strong operator topology to the solutions of the infinite-dimensional RDEs. Numerical experiments show the performance of the proposed methods.

Path Connected Components in the Spaces of Weighted Composition Operators with the Strong Operator Topology II

The path connected components are determined in the space of weighted composition operators on the space of bounded harmonic functions with the strong operator topology.

Iterated Aluthge transforms of composition operators on H2

In this paper, we study various properties of the iterated Aluthge transforms of the composition operators Cφ and Cσ where φ(z) = az + (1 - a) and [Formula: see text] for 0 < a < 1. We express the iterated Aluthge transforms [Formula: see text] and [Formula: see text] as weighted composition operators with linear fractional symbols. As a corollary, we prove that [Formula: see text] and [Formula: see text] are not quasinormal but binormal. In addition, we show that [Formula: see text] and [Formula: see text] are quasisimilar for all non-negative integers n and m. Finally, we show that [Formula: see text] and [Formula: see text] converge to normal operators in the strong operator topology.

ON ITERATED POWERS OF POSITIVE DEFINITE FUNCTIONS

We prove that if ${\it\rho}$ is an irreducible positive definite function in the Fourier–Stieltjes algebra $B(G)$ of a locally compact group $G$ with $\Vert {\it\rho}\Vert _{B(G)}=1$, then the iterated powers $({\it\rho}^{n})$ as a sequence of unital completely positive maps on the group $C^{\ast }$-algebra converge to zero in the strong operator topology.

UNCONDITIONAL CONVERGENCE IN THE STRONG OPERATOR TOPOLOGY AND ℓ∞

In this paper we study non-complemented spaces of operators and the embeddability of ℓ∞ in the spaces of operators L(X, Y), K(X, Y) and Kw*(X*, Y). Results of Bator and Lewis [2, 3] (Bull. Pol. Acad. Sci. Math.50(4) (2002), 413–416; Bull. Pol. Acad. Sci. Math.549(1) (2006), 63–73), Emmanuele [8–10] (J. Funct. Anal.99 (1991), 125–130; Math. Proc. Camb. Phil. Soc.111 (1992), 331–335; Atti. Sem. Mat. Fis. Univ. Modena42(1) (1994), 123–133), Feder [11] (Canad. Math. Bull.25 (1982), 78–81) and Kalton [16] (Math. Ann.208 (1974), 267–278), are generalised. A vector measure result is used to study the complementation of the spaces W(X, Y) and K(X, Y) in the space L(X, Y), as well as the complementation of K(X, Y) in W(X, Y). A fundamental result of Drewnowski [7] (Math. Proc. Camb. Phil. Soc. 108 (1990), 523–526) is used to establish a result for operator-valued measures, from which we obtain as corollaries the Vitali–Hahn–Saks–Nikodym theorem, the Nikodym Boundedness theorem and a Banach space version of the Phillips Lemma.